This is a good idea! Gaining advantage to Opportunity Attacks is less exploitable than gaining an Opportunity Attack when a creature engages you (and you're unengaged), and the idea is in line with the current way reach weapon works. But also, it doesn't consider the factor that, on the grid, you can melee attack with these weapons without being adjacent, so maybe only gaining this advantage to OAs is actually too little?
Let me think about that a little more!
Ah, this was considered! There are two points to this:
1. When Engage is a Bonus Action, melee classes which basically have little use for Bonus Actions could spend them round after round in order to "collect" engagements along the battlefield. Eventually, they would be Engaged with everybody, which is OP.
2. Actually, you can Engage for free as part of your movement if you're unengaged (which kind mimics the grid, where you move and become adjacent). It's only when you want to Engage an additional creature without taking OAs that you have to spend an Action. And this is usually only done when you have strong tactical reasons to do so: you trade an Action for the ability to impose disadvantage to all ranged attacks and spell of a creature, and to limit it's ability to Engage someone else freely or Dashing away. On many situations, it's very worth it!
Considering many 5e combats are 3 rounds, giving up attacks just to engage 1 guy is really really situational. Normally you would just kill them, after all...death is the ultimate condition.
Since your trying to make this system as tight as possible, it might be worth just dropping the action entirely.