Opportunity attacks have their long way in the world of tabletop games and D&D in particular. 3.5e/PF both had an exhaustive list of actions which can or can not trigger an OA.

5th edition simplified things a lot. One single trigger left:

You can make an opportunity attack when a hostile creature that you can see moves out of your reach.

It seems the developers consider this particular trigger as the most important one. To my knowledge, many wargames uses the similar rule. So what happens if we remove it?

Is there a substantial problem with turn-based gameplay, which OAs solve?

The reason I ask is because there are 5e-based games which does not have OAs at all (Five Torches Deep, for instance), so I want to know what changes should I expect within a basic 5e gameplay (no feats, no variant rules) if the DM introduces "no opportunity attacks" house rule.

I'm more interested in base mechanics, rather than particular OA-dependent 5e spells or features (like Rogue's Cunning Action becomes less useful, etc.)