Regarding your own experiment. From what I understand, the expertise bonus is always a d6 at any tier. The improvement comes from the number of times per day the d6 gets added. Perhaps, the character gains one expertise die per number of the proficiency bonus? Like hit dice, the expertise dice renew at the end of each long rest.We are right now experimenting with a similar system, where you can roll a limited number of d6s (daily resource) to add to a d20 roll. The rule is, it literally increases what the dice counts as, and like a true d20, it cannot roll higher than 20.
In this case, the expertise dice are almost identical to the Bard inspiration dice. As such, I dont mind its total with the d20 exceeding 20 − as long as it cannot stack with Bard inspiration. In other words, Bard inspiration is specifically granting an expertise bonus type. Likewise Bless, and similar bonuses.
With regard to all of these good suggestions for how to handle expertise. The rule of thumb always boils down to combat. What does ‘longsword expertise’ look like. ... And do the players mind when the monsters use ‘longsword expertise’ against them?