Pretty much the reason why a lot of progress in manned spaceflight came to a halt is due to the Shuttle program. It was sold on a laundry list of promises that somehow a lot of powerful folks utterly bought into. But in the end it ended up being far less capable and far more expensive than just about every alternative. Although admittedly it does look pretty cool.
It wouldn't be so bad if the orbiter didn't have such big wings (due to a military requirement for a large cross-range flight ability that was never actually used). Big wings means a more difficult reentry scenario, which means exotic, and brittle, thermal protection systems on the wing leading edges, which means a much larger area of highly vulnerable thermal protection materials on the orbiter. And it wouldn't be so bad if the vehicle didn't use liquid Hydrogen, which is super-cryogenic, requires excessive thermal insulation, and is extremely prone to formation of ice. A lot of people think that we were unlucky with the loss of Challenger and especially with the loss of Columbia, but in truth it was the opposite, we had gotten extraordinarily lucky prior to that. In reality the chances were very high that we would have lost a vehicle to either of those failure modes very much sooner.
But yes, it's a very troublesome design for a lot of reasons.
It's not really a question of wings so much as weight. The real issue with putting something on top is that what's below needs to carry that weight which is easier to avoid if the shuttle has it's own engines and sit's on the side. More importantly if the shuttle was not designed to send a few tones of stuff to orbit it could use the same basic design without experiencing anywhere near the same thermal stress and having a lot more safety margin.
Again, the thermal protection system was delicate because of weight issues. If the shuttles design goal was to get 20 people to LEO safely and they had anywhere close to the same budget to work with they could have used a few inches of titanium as part of the thermal protection system vs just glued on tiles that are less dense than Styrofoam.
PS: The surface area to weight is directly related to reentry heating. A person can do reentry in little more than one of those old style space suits and a parachute, the shuttle needed something that was barely possible to build.
The wings WERE a military requirement but nothing quite that silly.
It was to enable a launch from Edwards AFB in California, insert or extract a polar orbit payload and land on US soil.
Importantly it allowed them to choose the orbital insert point while they were over US territory where nobody was watching precisely where it went.
Silly? It's a well known design requirement - the shuttle has to be able to enter the atmosphere, slow down and deliver a nuclear bomb over any target on Earth (e.g. Moscow).
Looks cool? You only need to look at it alongside the awesome Saturn rocket to see which is inferior. It is clearly the one involving assorted cans held together with the rubber bands.
Pretty much the reason why a lot of progress in manned spaceflight came to a halt is due to the Shuttle program. It was sold on a laundry list of promises that somehow a lot of powerful folks utterly bought into. But in the end it ended up being far less capable and far more expensive than just about every alternative. Although admittedly it does look pretty cool.