I wonder if they even fixed Galen's flawed reactor design, and just patched up the exhaust port route and added some kind of regulator. So it took more than one shot to kill the Second Death Star...but it was still just two craft and a few torpedoes at the regulator and some missiles at the reactor, and the results were the same....just had to get the ships to the core directly.
Which would have been physically impossible had the superstructure been completed. Again; Palpatine's overconfidence was his weakness. Had he waited a year or two or however long it would take, the rebellion likely would have been crushed. But he had to leak the location early to make it look good when fleet showed up. Arguably he was also motivated by fear of how powerful Luke was becoming, how much of a liability Vader was turning out to be, and how widespread the rebellion had gotten.
As for it just taking two ships and a few torpedoes: I challenged anyone to design a high energy reactor capable of vaporising planets that can also survive having weapons grade warheads blowing apart it internals. That's not an engineering flaw, that's just physics.
That's been my theory since Rogue One; the Empire patched up the obvious flaw with the exhaust system, but either they never lost faith in Galen's work since everyone who knew him well died within the week of his not-being-broken-ness was revealed to Tarkin and Krennic, or Galen overestimated the capability and diligence of the Empire's other engineers and scientists and they genuinely never realized it wasn't an intrinsic fact of the Death Star reactor concept that it would catastrophically explode in response to even a small amount of damage. Either way, they didn't design a safer reactor for the Death Star II, they just added an external "power regulator" that, I assume, was supposed to keep the reactor from overloading and exploding if damaged, and didn't go any good at all if it itself was damaged before the reactor.
Again, the exhaust system was never the problem. Also it's not like it's the only thermal exhaust (it's specifically referred to as "a small TEP" and that it's "right below the main port") meaning there were other (probably many) small TEPs and the main port is sat atop the north pole and is gigantic (and can be glimpsed in RO.) That port just happened to have a direct line right to the main reactor, and probably for good reason.
So they really don't need to mess with it, just track down and fix whatever made the system to unstable. Which is probably what the function of the power regulator that Wedge took out was.
One of the officers reports to Tarkin that they figured out what the Rebels were trying to do and that they might be able to do some some serious damage ("Evacuate?! In our moment of triumph? I think you overestimate their chances."). Presumably, the Death Star was continuously reporting in telemetry and live-feeds, so headquarters would've gotten the same report.
Yup, and thanks to the novel 'Lost Stars' we even know it was a young officer named Jude Edivon that figured out what they were trying to do from the initial runs by Gold Squadron. Of course she didn't figure out what Galen had done, just identified the exhaust port as the target and reasoned that they must have found something in the plans to make them think this was a target worth throwing everything the had left at.
So far as I'm aware it's never mentioned in any story whether or not they Imperials actually tracked down the flaw and fixed it (though several reference books probably feature conflicting accounts, which is typical.) It's entirely possible all they did was apply so kind of brute force patch to make things more stable or protected from explosions and called it a day.