Think of it this way, Surprise (capital intended) in 5e models not even knowing a fight has started. It's a very specific thing, not just what we'd generally ascribe to surprise. That surprise is what is represented by advantage -- the surprise from having someone you didn't know was there (but you were still wary) attacking you from hiding, or the surprise you might feel when someone you're looking at suddenly lunges for you. The Surprise in 5e represents a very narrow condition, and one that does recreate itself once people are in a fight.
I mean, look at what Surprise does -- you don't act on your first initiative and can't react before that. That's it. Being Surprised doesn't even grant advantage. The confusion comes when we start talking about Surprise in reference to a class that has specific class features that leverage Surprise. But, it's important to note that this leverages that special status of Surprise, not surprise the natural language word. That you can do something surprising doesn't make it Surprise. And, outside of that class, the distinction between surprise (like hidden attacker) or Surprise is pretty minimal.