The risk is also that the same fighter who takes Level 16 of Cleric only gets first level magic that's useless against the 16th-level undead he's fighting.
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The problem can be partially solved like this: you have a character level that measures your overall vertical power. Your assortment of class levels measures your overall horizontal power.
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The above illustrates one of the more critical problems with this: you loose one of your main motivators for gaining levels.
(...)
So there needs to be some refinement of that idea.
I think this post was very good!
For me the idea that a 15th level Fighter taking a level of Wizard should get a 16th level Wizard ability goes against my suspension of disbelief big time, and it's very much a dealbreaker (just for using multiclassing in the game, not for 5e as a whole!).
It's the old example of "pianist surgeon" IRL. There is nothing in the world that prevents an experienced surgeon to pick up piano lessons, or a professional pianist to enroll to a medical school.
But the 15th level surgeon just can't pretend to suddenly play piano at the same level of the professional pianist,
even if all the surgeon can play is ONE song. And the professional pianist just cannot be able to suddenly perform a surgery at professional level,
even if it's just one specific surgery. They both can try something but their results just cannot be quickly brought up to a professional level, by a large margin.
So rather than sacrificing reason and believability for "balance", let's go back to your first sentence (highlighted) and see that what the 5e designers really need to do, is
make that 1st level magic still useful at 16th level.
The problem is really only with damaging spell, but if gamers keep reasoning only with "damage output" in mind, this is going nowhere. A Fighter should not be entitled to suddenly pick up a 100-dmg spell because it's plain ridiculous. But if the same Fighter picks up Burning Hands, this has only a very minor impact against a 16th-level monster. So what? It just means that this choice is a bad strategic choice for a Fighter, and what is wrong with having some bad strategic choices in the game? Why does every conceivable strategic choice always be as good as others? Trying to be a professional surgeon and a professional pianist is not normally a good life strategy (tho there have been rare people successful at both, but not starting their second career after 20 years in the first) why should it be so in D&D?
But OTOH is said Fighter picks other spells that by their nature are not damage-dealers but useful for other circumstances, there are actually dozens of valid options. Or he can even indeed pick up Burning Hands, and use it against minions only (which we know will still be a threat at high levels in 5e, so it's not a moot point...).