Opportunity attacks have a long history 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.

Various games have their own OA mechanics, but one trigger always remains the same.

• 5th edition simplified things a lot. Only the one single trigger left:

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

• Starfinder also reduces the list of AoO triggers greatly, comparing to its ancestor, to three points only, and "an enemy goes away" trigger is still there:

  When you threaten a space and the opponent moves out of that space in any way other than a guarded step

• Open Legend, a game influenced by but not directly derived from D&D, has the same only one trigger:

  If you are wielding a melee weapon, and an enemy moves from a space within your reach to a space that is not within your reach, you may make a free attack against the enemy.

It seems the game 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? Why do we need it, in the first place?

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

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