That has incredibly bizarre consequences, though. Let's say you're in such a situation where you've attracted attention and have so many turns until the whole site gets alarmed. In that case, if you wait, you can encounter people every single turn, but if you move around, you don't... so standing still amounts to a "shout really loudly and attract someone's attention right now" button as opposed to, well, waiting. Basically, with what's being suggested here, you have more time to kill if you wander around the place (which doesn't attract attention all that often) and waiting in the same spot draws attention immediately and on every turn, which is... rather counter-intuitive.
I would agree with the proposed change if time didn't really mean anything on a site, but in that case, it does. So I'd suggest that instead of making "wait" spawn an encounter immediately, just put in some kind of command that allows the player to wait multiple turns or until an encounter occurs.
You're pointing out that if a conservative has glanced at you, you can have more encounters before "conservatives suspicious" comes up if you hit wait than if you walk around. I don't think this is even noticeable, let alone incredibly bizarre. Time is abstract in sitemode. What does wait mean? Wait for one unit of time? Wait for something notable to happen? There's no purpose to the former, so we make it the latter. You could say it's bizarre that the
secret timer the player can't see doesn't tick more than once, but it's not like the player is going to notice the difference. If you saw the code and wondered about it, I could just say that standing in one place and looking innocuous while others walk by attracts less attention than moving around, and that would be the end of it.
Furthermore, why would this ever come up as an issue in real gameplay? Is there any reason at all in the game to wait pre-alarm besides wanting a group to show up? What is the player doing such that they have reason to wait, and don't want to attract attention? If you're looking for encounters, is there a balance issue that makes it a problem for you to get extra encounters? Is it even thematically strange that if you're looking for encounters, you can find more in a short amount of time than you could otherwise?
I have strong preference against further complicating the interface of an already overcrowded screen, especially when the added function is just a shortcut, not a new feature. I would rather remove the timer before "conservatives suspicious" comes up, and make it only possible to trigger it by getting one too many suspicious glances, than add yet another button just to wait for an encounter. I really don't think it matters, though. Making wait spawn an encounter only removes the ability to do nothing for a turn, which has no purpose in the game except to make your fingers tired... or to wait for a fire to die down in your exit path, or for siege units to move out of the way, both of which are post-alarm.