Wouldn't that be because immigration violations are ongoing events?

No, it's because immigration issues are civil and thus not easily amenable to 8th amendment challenges. Although some kinds of immigration violations are criminal acts, they're not considered ongoing criminal events for judicial purposes.

Edit: IANAL. You want to be very wary of any blanket statements on immigration law in particular, which is more complicated than almost any other field besides tax law. Like, I can show you decisions where the Chief Justice of the Supreme court simply makes inaccurate statements about the law by mistake. And others that seem to contradict what I've said above (but where the contradiction is in dicta rather than being part of the holding).

Well, even if you leave, if you've been present illegally in the US for more than a year, you're subject to the lifetime bar. So, no, they really don't have a statute of limitations.

That's inaccurate. Illegal presence greater than 6 months and less than a year subjects a departing alien to a 3 year bar on re-entry absent a waiver. Illegal presence for longer than a year ups that to a 10 year bar. There are also lifetime bars on re-entry but they are reserved for much narrower and more egregious violations, like election fraud, aiding terrorism and so on.

You are correct, my recollection was faulty. However, the lifetime bar is not that narrow. It applies if you are present illegally for more than a year (or are ordered deported regardless of length of stay) and then leave and (attempt to) re-enter illegally again. So anyone long-term present in the country illegally who has ever left and come back would in principle be subject to this. That is at least my impression, but I'm only an immigrant, not an immigration lawyer...

That's Ohio, so there's very little immigration related law there.