Thought on banning Multiclassing?

I'd flair this as 5e / 5.5e if I could, but oh well.

I've been compiling a big list of my houserules and noticed that a large portion of them could be solved via banning multiclassing (side note here: yes, in 5e multiclassing is an optional rule, therefore a GM is always free to just "not let it through", but it was so widely accepted that it was presumed it was on the table at all times; in 5.5e, however, it has lost that moniker, so now it's considered more of a baseline that has to actively be banned. I'll be using the word "ban" from here on just to save time), and that got me thinking: have others done that and how did it work out?

From my PoV, the pros are the following:

1. Removes cheese builds, the most obvious being people who take level 1 in Fighter and then go X levels of a caster class, since that allows a character to have proficiency in heavy armor and still be able to cast very heavy-hitting spells. So instead of having a glass cannon Wizard or Sorcerer, you have... a standard, metal cannon.
2. Allows for more creative homebrew dedicated to each class without worrying about things like "BuT wHaT iF tHeY mUlTiClAsS tHiS aNd ThAt and break my shit?" I'd really love to give a lot of love to a few classes I personally deem as a tad subpar, but multiclassing always seems to rear its ugly head and kill my enjoyment for that sort of stuff.
3. Possibly making more feats way more appealing to take. I myself would start looking at things like Lightly Armored if I couldn't just 1 lvl dip into Fighter to get up to medium armor for basically free.

And, again, from my PoV, the cons are:

1. Banning is kinda boring? But so is cheesing the game, so they probably cancel out.
2. I know that sometimes you NEED to multiclass in order to attain a certain flavour fantasy you might want for your character (i.e., a Paladin / Warlock "Witch Knight" archetype; think Arthas from Warcraft III), so outright saying my players can't do that in my games feels just outright bad.

So I'm looking for advice on the topic. Oh and spare me the "just make it so a multiclass needs to narratively make it sense", because it's comically easy to come up with an explanation for any multiclass combination. I need to make it a hardset rule: allow or disallow, no ifs or buts.

Thanks in advance!