Multi-classing into a class - any class - should IMO never provide all the same benefits and features available to a single-class, or at least not as easily or early.
Why?
To discourage dips based on nothing other than optimization, to discourage jack-of-all-trades characters (a group of specialists need to be more of a team than a group where each character can do a bit of everything), and to - very intentionally - make multiclassing just a bit less attractive. (and with all the classes and sub-classes the game has, odds are that pretty much any not-OP character concept already has a class ready-made to suit it)
There could be an entire discussion on how we change the multi-classing rules, but again, I see that as an entirely separate conversation than buffing warriors. And since I have yet to meet a single person who has problems with warriors multi-classing to get spellcaster abilities, I don't see much of an issue with the inverse.
I don't care to make rules specifically to prevent someone from doing something just because you might consider them doing it for the "wrong reason"
I see nothing wrong with making jack-of-all-trade characters, nor would I care to make rules to prevent a character archetype because of your ideal party composition doesn't include it.
And while I don't like multi-classing, I don't want to make it lesser for people who do.
My idea for the weapon mastery bit would keep them down for a full round at cost of doing any damage, i.e a change to how it works otherwise.
Ah, that was unclear.
I could see making that an option, but some of the masteries wouldn't have the ability to deal zero damage, due to their nature. I think something like that would be better fit for a feat, otherwise balancing the masteries would be incredibly difficult.
Given that in every survey anywhere Fighter remains the most-played class, how commonly-seen in the wild are all-caster parties?
Pretty common considering over half of all classes (9 out of 13) are casters or half casters. The thing is, everyone thinks you need a fighter or a barbarian to tank, or a rogue to deal with traps, but you don't. It is an issue of perception.
For people who want to build a "balanced" party, going Bard, Paladin, Wizard, Cleric is incredibly powerful with no obvious downsides. But going Fighter, Monk, Barbarian, Rogue leaves clear and obvious holes in the group that need to be filled. Both should be equally viable, but they actually aren't.
There was a very informative set of videos from the Dungeon Dudes that highlighted this for me. They break roles in the party into seven categories. They then ranked every class in how they handled each category. To summarize?
Fighter and Barbarian were ranked the lowest possible in 5 of the 7 categories. The only things they were good at were Taking and dealing damage. But most other classes were better than them in those five categories. They weren't alone in some of those rankings, Rangers make terrible negotiators, Rogues are bad at support But most classes had four categories they were one of the BEST in, the few that didn't (Cleric, Ranger, Sorcerer, Warlock) were solid mid choices for multiple roles. The exception being the Monk which fit poorly into most roles.
I know you are old school, but even you have to admit that if given a choice between someone who can guarantee a success in something or someone who has to bend and twist and hope to be passable in it, the better choice is the guarantee. And what we have seen is that the three Warrior classes (Barbarian, Fighter, Monk) are the ones that are the least flexible and the least able to fill roles outside of "damage". And they aren't the undisputed best at those roles.