Sims definitely do visits. At least from my style of play. As a D2 A+ school, I make a living off pushing down D1 sims. When doing that, I'll begin with promises. If the promises get me to moderate (my RS1 peak) AND bump sim down a notch (say, to high), lets say I add 3 visits next cycle. That might only bump sim to low when it processes.
If I don't add more visits, and use only AP the next cycle, the sim will jump back up in one cycle. As far up as very high again, depending on the situation. 27ish AP from a D prestige, D1 sim wouldn't make them jump 4 levels. Not against an A+ D2 school that has visits and promises offered. That would be ridiculous in my opinion. I've estimated that when you fight sims from D2, they do about 2-3 HV each cycle during pursuit. And that continues until either A) you run out of money or stop pursuit. Or B) you push them to very low.
Add on..... some will say that if you push sim to just low, they will quit. I've learned a LOT about this situation lately..... in some cases it will "appear" that sim is dead at low. But they aren't. Its more of a "pause". While targeting a sim led PG one season I pushed sim down to low and wasnt willing to spend anymore on the player. I expected the sim to rise back to high next cycle, but they didn't. But while snooping around, I noticed the sim that was involved with my player, was also very high on another PG. So I think the sim will prioritize their situation, ONCE THEY ARE PUSHED DOWN TO LOW.
In my case, the sim team was Umass. And they held at low for many cycles without me adding any effort at all. Then Umass won the roll for the OTHER PG they were pursuing. At that point, Umass was out of the running for my PG. Had UMass lost that other battle, they likely would've switched back to my target and picked up the pursuit, since I never pushed them to very low.
Point is, I've found out how other things can factor in a battle between you and a sim. It's not always just about what you're doing. It also involves what sim is doing elsewhere. And there's things you do not see, that may lead you to believe that you're safe when you're really not. Always push sim to very low if you want the player!