Wow... That title...
It's about expectations and mutual respect. You're playing a game together: the players and the DM, each person has it's own responsibility, but there is a shared ownership and not an absolute ownership. Not for the DM and not for the players. But if people want to and mutually agree that you want to play in an extreme adversarial game, have fun in that.
It's simple: If people don't have fun, they'll stop playing and everyone loses. That's true for the players
and the DM. It's generally about meeting somewhere in the middle. If someone wants to be an adhoc wainwright, why stop that person from doing that? It's something different if someone misuses that constantly and is being a Jack of all Trades and Master of All just based off the PCs Int score, while min-maxing the heck out of their combat skills. You know the type... There are mechanical options for that in the game if you want to do something with that, just depending on ability scores is imho a little lazy. On the other hand, if the PC is spending their downtime constantly learning about wainwrighting, why not reward that with a tool proficiency?
If you can't give a little if you're a DM, you shouldn't be a DM imho. But you also don't have to dance to every player's whim. A simple solution is: Yes, you figure out how to put the wheel back together again, but if you actually want to repair the wheel, that's not not an int check, that's a Dex check without proficiency and if you don't have the appropriate tools, that's minus X. But you're a wizard, if you were interested in fixing stuff you would have taken Mending (1st level spell)... I know how a ball should go into a goal in the game of soccer, but knowing how it should be done and actually doing it are two
very different things... Maybe that is also true for DMing?
And if a DM is being an arse about it... We have: ChatGPT please write a detailed 200 page background story for a wizard, please include wainwrighting skills in there somewhere...