And it will cut down on the cheese: 1st level fighter, then wizard X
that's not "cutting down on the cheese" - it's giving up entirely and just integrating the cheese directly into the core rules. which, i mean, if you don't care about that then ok, but that is what it is.
and honestly i think you could get away with doing the exact opposite (if you wanted to change this at all) - split weapons into categories based on what they are instead of the level of training needed to use them (e.g. swords, axes, polearms (edit - actually maybe spears instead, and make 'polearm' a property that triggers feats like polearm master? or maybe just classify 'polearms' as a weapon of any category that has reach? either i think would be more useful then a polearm category since, for example, poleaxes are both polearms and axes), hammers, picks, bows, crossbows, etc.), maybe add a "simple" trait for weapons any character can use proficiently (like clubs), and then give classes proficiencies based on that (e.g. rogues get swords and short weapons, bards only get swords, barbarians get axes, hammers, and swords, rangers get swords, bows, and crossbows, fighters pick any 3 or 4 categories, etc.). makes it easier to assign proficiencies to classes and gives each class more character in what weapon proficiencies they get since it's more specific. of course, that doesn't matter if the weapons are as samey as they are in 5e anyway, but that's another (and i think the actual) issue. also,
yeah you could make 4 basic attacks like cantrips and let them choose 2... then you can have 4 sr/encounter ones and let them choose 1 and a daily ability too... and every few levels they can get more sr/encounter or more dailies...
4th Edition strikes again!