It wasn’t ‘stupidly wrong’. Garret Reismann is their technical advisor, and he was asked about it. It came down to a budget problem - they had enough money for a moon base set, or for a new ticket design. Using a Shuttle meant they could use stock footage, and there were sets already built for the interior. Shuttle space suits were also available.
So, they fanwanked that by mentioning on the show that the Shuttle had to refuel in orbit to get to the moon. And that in this alt timeline where there was a moonbasd it may have had other changes like radiation hardened avionics and such. It still wouldn’t work, but they did the best they could. Also, using the shuttle added a connection between the timelines with a spaceship the audience was familiar with.
It’s not hard to imagine that future with a race to a moon base, the requirements of the shuttle might have included ability to go to the Moon. It would have been a very different vehicle, but close enough.
I can accept deviations brought in by the reality of filmmaking. Another example was that it would have been really hard to show astronauts constantly in 1/6g, so they just occasionally showed something to remind viewers of low gravity. That’s just the reality of medium budget filmmaking.
But for an example of how hard they worked for accuracy when they could, the original mission control set was so accurate that they went to the trouble to match the correct paint on the door hinges. The layout was not only the same down to the inch, but every console showed the proper display for that console for each part of a flight. The technical manuals at the consoles were the actual ones they would have used, etc.