I don't know what it is with this push in D&D that everything has to be unique or available to everyone, as if there was no middle ground. Classes cover a range of abilities, dividing the roles, niches, and responsibilities of the various D&D genres among them. There is overlap, and there is distinction.
I agree. I only quote because in terms of the Melee Combat (ie swing a weapon, hit/miss/do damage), everyone already has access. Spellcasters get a monopoly on spells i.e. wizards can always swing a weapon, but fighters can't ever cast a simple spell (class only here, ignoring multiclass, specialties, or other "add-ons" here).
In this regard, expertise dice represent a *better* (ymmv) approach to Melee Combat and should be adopted "universally" (if you agree its better). Does this mean that wizards should get expertise dice? No. They get the lame basic attack with weapon. Suck it spellcasters!
In this viewpoint, expertise dice are an analog to spellcasting. Each attack can be unique and flavorful just like spells are unique and flavorful, and spellcasters can be locked out.
I get the issue people have with fighters not having a unique "schtick". But with expertise dice, the problem is narrowed and rather than look at fighter Vs. spellcaster, we are comparing fighter Vs. rogue. I think that's a step in the right direction. Just as Clerics and Wizards are different despite both being spellcasters, we still need a clear, sharp line between Rogue and Fighter despite both being "Expertise users".
Personally, I would do something like:
Rogues have less expertise dice than fighters (since rogues get more skills)
Rogues can apply expertise to skill rolls.
Rogues can only use expertise in combat when they have Advantage.
Fighters have the most expertise dice.
Fighters can use expertise dice any time (including on anothers turn, see below)
There would be Basic Maneuvers (available to all with expertise dice), Fighter only maneuvers (class features) - Many of these would be reactions like 4E to help allies, block attacks, etc
Rogue only maneuvers (class feature) - Many of these would be for added movement such as hide, or move through enemy squares, etc.
This is not well-thought out, but just a brainstorm of what is possible.