How would they proceed with that knowledge that they couldn't possibly know?
The USS Hiraga was 2 weeks away at maximum warp when they first got in touch with the Khi'eth letting them know they were mounting a rescue mission. Which means the Hiraga knew of Khi'eth's conditions, what kind of ship it was, what was the nature of its mission, and the kind of nebula they went into.
When the Hiraga arrived at the nebula, they should have conducted a more thorough analysis of it to see if they need to make further modifications to see if they can survive the trip through the nebula, and if not, hold the position without entering the nebula and consult with Starfleet... and if they just went inside the nebula without any kind of regard and experienced similar problems Discovery did, why not retreat to outside the nebula? They would have had time to exit it at impulse like Booker's ship did.
I don't know why. I would ask the Starfleet Engineers.
Bottom line is: we don't know whether what we saw was supposed to represent the TARDIS tech or if it was a VFX mistake.
Most people are treating it as a mistake, not as TARDIS tech. It bears addressing whether the technology was integrated in the first place (so at least it would give maybe Disco crew ideas on how to make better use of it - and make them even more useful than 32nd century trained SF officers and engineers).
Also, I would ask SF engineers, but sadly, since they are fictional, I guess the writers would be a good place to start (and fat chance we'll get a shot at that either - but if I did, I would be asking the questions with genuine curiosity... not bereavement).
How can they rise above mistakes they haven't made? Writing is an art; I don't think I should have to say that but I will. These writers didn't work on past Treks so how are they to know all the past mistakes? What makes that possible?
The internet does (less than 5 minutes search needed to see what previous technologies were used, what was said about them, etc. - they conveniently forgot that there were 2 versions of Slipstream for example, one that never needed Benamite crystals (and allowed 300 Ly's per hour), and the second one which did (and that the technology to synthesize them existed in the late 24th century - with R&D and hundreds of years later, it would be a piece of cake to make new ones - in fact, anyone with programmable matter could likely make them, and that technology seem to be abundantly pervasive in the 32nd century despite the Burn). Then there's the fact they have people working on the show who are supposed avid 'fans of Trek' and even claimed to have 'Trek science advisors'.
So, these would be just some of the reasons I'd expect higher quality writing and the show not repeating past Trek show's mistakes. Its one thing if they make an honest mistake every once in a while, but to do it blatantly with every single episode? That's just indicative of lazy writing.
I don't know anything about the Expanse so I cannot comment. I'm going to guess though that going that way would largely having massive pushback because it isn't Trek enough, given the response to Discovery in some corners in the first place. So, the writing staff has basically been sufficently cowed. They won't advance too much, but they'll do just enough now to play it safe. The main character is now a captain, she will have various advisers who will help her solve problems of the week and they will warp off/jump away to new adventures. Lather, rinse repeat.
What wouldn't be Trek enough?
The writers knew that pushing the series so far ahead in the future would had to have come with massive (ridiculously larger) technological changes and so many other changes on a galactic level. Pushing the show 930 years beyond mid 23rd century and keeping it in the same universe with most data intact (like SF was shown to have retained) doesn't give you the excuse to ignore what happened before. You need to build on that... and need to do some decent research on the topic (which is incidentally what any good writer would do) to cover up majority of blind spots that could occur and plan ahead. They have 'teams of writers' for crying out loud... not just 1 person... how the heck can these people make so many freaking mistakes that are so 'basic'?
Its like being given the ability to use the internet (and past Trek shows scripts) for research and then deciding not to use it 'because of reasons'.
They could have also drawn from real life hypotheticals on what Type III and Type IV civilizations (on Kardashev scale) would look like and transposed that on Trek's 32nd century for example (which is what I mentioned before).
Transwarp beaming is yet another 'nugget' that was never realized (that alone would have bypassed the Burn easily enough), as are alternative methods of power generation.
In short, 23rd and 24th century Trek gave the writers PLENTY to work with to create a much more advanced setting compared to what we saw that was convincing and structure the stories around it - missed the ball on that one.
Again, we may see some of this in Seasons 4 and 5, but I'm not so sure after what we saw in S3.