I think restricting tripping (and similar effects) to a couple of sub classes is part of what makes 5e combat a bit stale. I've never seen any character other than a Battlemaster or a monk try to trip someone. Opening up maneuvers for others will enrich combat. Battlemaster's can still shine as they can stack an Advantage effect with their maneuvers for potentially powerful combos.
But if anyone can do what a Battlemaster or Monk can do, why be a Battlemaster or Monk?
Here's another idea for expanding Advantage / Disadvantage: Resistance to damage means you roll damage dice with Disadvantage; Vulnerability means you roll damage dice with Advantage. Critical hit? Roll with Advantage.