I generally view most concerns about balance with either option as wildly overblown, and probably won't enjoy playing with a DM who is banning either option.
If they ban if for balance reasons, they probably don't run a game that just lets the PCs do cool stuff and enjoy their character, IME. Rules interpretation errs on the side of telling the PCs no, etc. Not something I'll enjoy, and I'll end up disrupting the game because I have limited willingness to just keep quiet and ignore something I strongly disagree with, especially when it's something I have a lot of experience with, like DMing.
If they ban it because "classes take years of training", I probably won't enjoy how they interpret things like what a class means in the fiction, or how literally you can take the interaction of mechanics with in-fiction physics, etc. I also just really dislike that POV on what classes are, and am not willing to restrict myself to it. If I want to use the rules for the Swashbuckler Rogue and Bladesinger Wizard to represent a specific school of swordfighting, and none of the options involved are banned in general, I am going to do so. Period. If I want to take a level of fighter at level 10 because it does a better job in my opinion of modeling the expertise I've acheived with the sword, then I will. In my own games, I don't consider decisions like this something the PCs need my permission for, or that I have a right to deny them.
I have asked my fellow DM's permission for build elements before, because I wanted to make sure he wouldn't be annoyed down the road by the gameplay they imply, like making a tinkerer alchemist that enchants magic items. Magic item crafting is world dependent, and invention has basically no rules for it in 5e, so that build asks a lot of the DM, and I'd have been happy to adjust the concept if he hadn't been up for it.
Me playing an Open Hand Monk/Moon Druid doesn't ask any more of the DM than playing a monk or a druid, so if I start talking about my concept and the DM starts telling me no, I'm probably going to find something else to do rather than waste both my time and that DM's time. Our outlook on TTRPGs are incompatible.
That all being said, I would play in a game without either option if the DM had a reasonable and interesting concept in mind and the lack of either option seemed to legitimately contribute to that, or if a DM I really trust and know well proposed it and asked me to just trust them.