Not necessarily. I've seen flights where they have the offers and don't end up needing them. It's all dependent on if they think they might need someone (or a few). I don't know for sure, but I imagine the offer comes up if they are overbooked even by a single person.

FYI...saying yes at check in just tells them you're interested and puts you on the list. Nothing gets finalized until the gate, after check in closes when they know who's checked in or not, have a better idea if/how many misconnects are likely, etc. Depending on the needs, they may give you a higher number later (they still give everyone the same amount, I think, so if on day of they solicit volunteers at the gate for $500, anyone who takes it would get that). It's still also contingent on you getting an alternative that works - i.e. if there are no seats on other flights that day, and you don't want to wait until the next day, you don't have to take it.

I don't know for sure, but there's also likely other factors they take into consideration - like if you have bags checked international (it's harder to pull those vs. just giving it to someone who doesn't have checked bags), the complexity of the itinerary/ease to rebook (i.e. someone just going to LHR is easier to rebook then someone with connections, especially on a partner)