Hiya!
How often does it "seem appropriate?"
Nearly every combat? roughly half of combats? Etc. Poll still seems to apply.
Do you play with theater of the mind only? With grid combat, AoOs are generally evident by player and monster action. The DM doesn't have to determine appropriateness.
1. How often?
I'd say...maybe 20% of the time a combat happens? It also almost always happens 'before' combat; like a total surprise where the target/victim has no clue another combatant is around. If/when it happens in combat, it's usually factored in because of a PC's "focus/goal" than any RAW specification.
2. Every combat?
Nope.
3. Roughly half of combats?
Nope.
4. Grid or no Grid?
No grid. Usually. Sometimes I'll whip out the whiteboard and draw out the 'area', then we use tokens for everyone/thing (Pro Tip: If you go to a hardware or big box store, those "floor-protector-stick-on-felt-pads" that go on the bottom of table legs and stuff?...They come in all sizes and shapes, and I use a Sharpie marker to list a number or a letter; PC's are Letters, monsters are Numbers; works really well!

).
5.With grid combat, AoOs are generally evident by player and monster action. The DM doesn't have to determine appropriateness.
Ahhh...and here is where my "It depends, as appropriate" needs a bit more explanation I guess.
I/we generally don't "use" the AoO as written in the game. I take what the game says (which generally tends towards the 'not seen' or otherwise 'if X, then Y'). For example: A PC is fighting an Orc. Another orc runs past his 'buddy' and the PC in order to get out the door and go for help. Normally, iirc, the RAW basically says the PC gets an AoO on the orc running past him. In my game, no...at least not without possible consequence. The way I see it, the orc engaged with the PC is looking for any opening in the PC's defenses in order to get a shot in...the PC is likewise doing the same. Another target (the running orc) is a VERY big "opening in the PC's defenses"; if that PC chooses to try and hack him to prevent him from getting to/out the door. My ruling in this situation is basically that the PC can either keep his focus on the original orc. Or he could take a swipe at the running orc, but if he does this, he is basically switching his focus...meaning his original orc target can get an AoO
on him as well (so PC gets AoO on running orc, other orc gets AoO on PC). The one running by doesn't get to attack anyone...he's trying to not get into combat.
So, as I said, "when appropriate". Usually my players tend towards not taking an AoO in this sort of situation.
However, in a situation where the original orc (not the running one) decides to follow his buddy, then the PC gets an AoO on him. Additionally, the PC can choose to immediately either run after the orc, OR he can get one more swing, but the orc then moves his full 'run distance' away from the PC.
Also, just to note, an AoO in my games doesn't use a "Bonus Action". An AoO in my game is 100% 'free' from the action economy...there are other drawbacks/factors/risks that take that 'economic cost' in a players decision. My DM'ing style and our general preference is more towards the "loose/flowing rules" than "tactical/RAW" style, obviously.
Hope that clears it up.

As I said..."As appropriate".
^_^
Paul L. Ming