Let's be honest. The Shenzhou was left floating out there like some big staged prop. Nothing more complicated than that, why try and make it logical in terms of the story.
Try centuries. Don Quixote is one of the earliest examples, while Huckleberry Finn is only the most famous. And then there's alot of the fiction based on/inspired by Guy Fawkes, which is both meant as a bit of antihero worship and gallows humor for what an asshole fawkes actually was.
Tomato, tomato. Simple resources are used for technology. Further their cause with scraps ? Yes, I remember all those times Germany used Coal from sunk U.S Navy Ships instead of its Coal Industry in the Saar. That's just absurd, you know. There can't be possibly that many destroyed Starfleet ships enough to "further the cause" of the Klingons. The only time we saw Klingons using scraps left behind by Starfleet it was by starving ones and the whole ordeal proved to be redundant. There's no precedent in the real world or Star Trek of the tide of wars being changed by stuff the enemy left behind. The only straight forward thing you have here is killing the people trying to escape for nothing. Oh, yeah! Because stuff being left behind in battlefields is not something realistic. Look at all these staged props!
As others have pointed out, scuttling a starship is an option for Captains to prevent the capture of their vessel and/or crew by the enemy. There are no standing orders requiring it, though, which is why we have only ever seen self destruct mechanisms being activated as a kind of desperation tactic when the bad guys are about to capture the ship and use it against them. Shenzhou, by comparison, is a floating hulk with no power, systems disabled, and unable to be restored in a reasonable amount of time. So much like the Constellation, the Exeter, the Stargazer, the Hathaway, the Jenolan, the Pegasus, the Lantree and even the Enterprise-D, the recovery of a wrecked starship is probably something best left to OTHER STARSHIPS which might eventually tow it to a scrapyard or a dry dock to be re-purposed. IOW, if it wasn't for the fact that the entire fleet was suddenly locked in a truly epic war with the Klingon Empire, standard procedure would be to take the Shenzhou in a tractor beam and tow it to the nearest starbase. Now that the war is basically over, that will probably end up happening anyway.
It looks like you're hedging: If it was not recovered, then the point that Starfleet values personal effects is meaningless. If they consider such more important, then the telescope was probably recovered. I don't see how crew members will bother to carry bulky objects containing personal effects (especially belonging to others) as they rush for escape pods. More likely they will consider anything portable that they think will be detrimental to Starfleet if recovered by the enemy first. As for recovery efforts, I think it is more logical for Starfleet to recover what can be used for the war effort and that might be advantageous for the enemy before anything else. Finally, I think the scene is a poor example of writing not only because of the plot hole but because it appears to be a rushed attempt at creating sentimentality.
It might be intended, but the idea that someone grabbed the telescope during the emergency evacuation into a lifeboat is ludicrous.
Lucky break really that Georgiou's last will and testament that bequeathed this telescope to Michael justified someone's telepathic efforts to retrieve it.
The Shenzhou isn't the the biggest issue with this plot element. Starfleet could have easily gotten their hands on the Klingon cloaking device. The writers obviously had no idea how to get out of this and chose to ignore this by moving six months ahead and thereby creating this huge plot hole.
That doesn't make any sense. If Starfleet values personal effects, then necessarily they have to board every battle-damaged starship floating around ? They can simply tell the crewmen they are allowed to bring onboard the escape pod what they think has sentimental value for them. Is that like rocket science ? Who said about carrying it for others ? I never said Starfleet demanded its crewmen to recover the personal effects of other crewmen. Besides, Jesus, this is the future. There's technology that's actually made for the whole purpose of carrying heavy objects around, you know ? I think Saru is a big boy and he can manage to carry that Telescope on his own. Evacuations are made in a rush but they do have a certain level of organization. Otherwise people would trample each other. I wouldn't doubt they have even one or two escape pods to spare on those fancy ships. How the hell people will not bother saving what's sentimental for them ? I'm not talking about bringing your pillows or your Playstation, I'm talking about the types of objects people bring on board to remind them the people they love back home. Ah, yes! All those darned secrets the Shenzhou carried around in its computers when she was exploring the galaxy and probably doing very exciting things like repairing damaged Federation relays. The whole tide of the war could've been changed only if the Klingons cracked the code of the Lateral Vector Transporter! I mean, come on, they specifically showed us these Starships like the Shenzhou were just regular, boring, exploration vessels with a bunch of people who had no experience in war before. There's nothing game changing on those portable devices you're talking about. I know DISCO looks like BSG but we're not talking about the Colonial Fleet here. Okay, what technologies common in the Federation are not available for the Klingons then ? Because in Discovery they made it clear the only stuff the Klingons don't possess is the Spore Drive. You can't just call small worthless details "plot holes", just because you're feeling pedantic about a TV Series. Plot holes are stuff that stop the flow of the story completely. I really doubt a 15-minutes sequence of Saru carrying that Telescope around would've told us anything new. There's stuff that is just common sense and there's no need to overly-explain it to your audience. They evacuated, they brought personal stuff along and that's the end of it. It's not complicated, it is only if you want it to be. That way I will start calling Saru's taste for blueberries a plot hole too. What the hell do they meant by that scene ? When we first saw Saru he didn't look like someone who ate blueberries. How did he acquired that taste ? I demand an hour long sequence explaining it. "I served with Georgiou god knows how many years, I'm very sad because of her passing, I can run like hell and I have superior physical strenght... But you know what ? Fuck all those personal belongings of my dear Captain. I'm outta here" - Saru, probably
It's a plot hole. They took the time to take a telescope with them but never considered to get the cloaking device from the Klingon ship. If they had shown that much intelligence we wouldn't have gotten the mid-season-finale with the dozens of spore-jumps to break through the cloak. And Lorca wouldn't have be able to gather all that data for the his mysterious final jump to only-Locra-knows-where.
To be honest, everything was staged. In fact every single Klingon and Starfleet officer was actually an actor. I think most of us are aware. By the same token, why try and look for problems? The story is actually pretty simple - the scene is left more or less as we last saw it. I'd say you're complicating it by arguing it's standard procedure to clean up battlegrounds and there must be a reason it didn't happen. Again the only reason anyone's even thinking about this is because a few stubborn Klingons sat around for months without being rescued and refusing to die and brought a dead ship back to life, which is totally unexpected both by us and in-story. Let's imagine this played out differently: the remaining crew of the derelict sarcophagus ship also launches escape pods, then appear on some other ship in episode 4. Then at some point the Discovery returns to the site of the battle and finds the sarcophagus still split in half and dead in space, along with all the other derelicts and debris, and Lorca salvages a dilithium whatsit or something from the Shenzhou. I'd bet you wouldn't bat an eye. So a single personal item surviving breaks your suspension of disbelief, but had they come home with the freaking Klingon cloaking device itself without any explanation as to how they got it, it would have solved it? Oookaaay... They left the Klingon ship without Georgiou's body to avoid further risk. Of course they aren't going to go back and poke around for a cloaking device that we don't even know if it's removable and able to tuck under your arm and carry to an escape pod. Whereas the telescope is sitting on a pedestal in a room just off the bridge. Come on now.
Why? To use it themselves? Because the Klingons weren't using it at that point either and I expect Starfleet didn't think of the device as a serious threat until all of them started using it. They probably assumed that T'Kuvma was a one-trick pony and that the secrets of the technology died with him. If Kol hadn't both salvaged the ship and found a way to reverse engineer the technology, they would have been right. An interesting question is why it took the Klingons six months to go back and salvage the device. THEY didn't think it mattered either, and it's possible that Kol is the only person in the galaxy who truly realized the cloak's potential (even T'kuvma seems not to have grasped the strategic significance of it). In a sense, it's sort of like military aviation technology. Even after the demonstrated power of aircraft to completely fuck up a battleship's day, the U.S. was slow to take the threat of naval air power seriously. It was a relatively simple modification of the Japanese long lance torpedo that allowed them to be so devastatingly effective at Pearl Harbor, and America didn't ween itself off the dependence on battleships until well into the war. Hindsight is 20/20
You see that doesn't make sense in story either. When Michael gave Saru the telescope he expressed (as much as Saru can) touching recognition it was Georgiou's. So this inseucre Officer takes the telescope in the first place, is magically contacted by God know's who, that it is a bequeathed item in a will. Lol. He then sends it off to them. It is then packed up with the recording, quickly finds new science specialist Michael (good thing she wasn't in prison it would have to be resent!). THEN when Michael gives Saru the telescope he gets to rest his sensitive gaze on something he had apparently not that long ago chucked into an escape pod. Nah. The telescope shtick was to squeeze out sentimentality. How it got there is best not defended.