Problem: Tracking the expenditure of 6 dice, round after round, is tricky. Sure, you could set up dice in front of you, but they'll be knocked around, or you'll forget to refresh, or the guy who just dropped a Horrid Wilting will pick up all your nicely arranged d8s.
Solution: Certainly no more than 4 dice in the pool, and 4 is probably pushing it. Solution only to be implemented if there is actual playtesting of mid-high levels. This probably actually means that we need to have sets of sets of dice, rather than just sets of dice (rather than having 12d6, you have 4 times 3d6).
Problem: IF multiclassing is done a la 3e, then being able to spend dice on extra actions/attacks can be balanced for low level fighters. It can be balanced for high level fighters through *really* big die sizes and putting sets of dice as a "die" (say, 4 sets of 4d10 or something). It cannot be balanced for high level non-fighters splashing fighter (because they are now spending 1d4 for an effect that is balanced for a Fighter at 4d10).
Solution: Either all effects depend on the size of a die-set (no extra attacks, and really, pretty limited options honestly) or all effects that are independent of the size of a die-set require significant investment in Fighter levels.
Problem: Rolling 10d6 once/fight (Fireball) isn't that big a hassle. Rolling 6d6 every round, in different combinations, along side dice of other sizes, is a hassle.
Solution: Forget dice, spread out "dice" in units of 5 (because adding units of 5 is relatively easy). So at level 1 you get 1 packet of 5. By level 5, that becomes two packets of 5, or maybe 1 packet of 5 and 1 packet of 10. Yes, dice are more visceral. Unfortunately, in practice, they'll be, at best, hovering at the edge of unplayable at high level.
Problem: Only 1 reaction per round.
Solution: No abilities that trigger out of your round, or Fighter Combat Superiority ignores the limit. Not that big a deal really.
Problem: If the dice pool refreshes at the start of your round, then any dice that get saved for out-of-action abilities that don't get triggered are lost.
Solution: Refresh the pool at the END of your round. Simple.