This is supported by the rules.None of this says when surprise ends. You can assume that it ends on a character's turn. However you can also assume it ends after the first round of combat.
You could also then say that surprise never ends, which would have more support for that interpretation because there are no surprise rounds.
I think it is much more reasonable to say that surprise ends when the consequences of the surprise have ended - that being not taking a turn and not being able to take reactions. If you are not affected by surprise, why are you still surprised?
There are really 2 things that we are talking about. The anatomy of a round and the assassin subclass.
Maybe this will help you to think of the assassin differently, I picture the design of the subclass to go something like this:
Jim: Assassins are a popular trope so we should include them.
Bob: Yeah, and we could use a subclass that works with attacking/sneak attacking. That sounds good for the assassin
Jim: What do rogues like to do? They like winning initiative.
Bob: Advantage would be a great reward for a rogue.
Jim: Okay great, if they win initiative then they get advantage.
Bob: Well wait a minute Jim, rogues like to hide and get the drop on enemies. Having an ability that gives advantage will make it so they don't care about doing that.
Jim: Okay, so they get auto-crits when they win initiative and their opponent is surprised. This will be the most common form of them having advantage in the first round.
Bob: Well autocrit is both very powerful, esp. for a rogue, and it devalues the fun of rolling a crit, but I suppose it will happen rarely and make that roll for initiative have more tension. Sounds good.
The key here is that both parts of the assassinate ability rely on winning initiative. That is what the ability is keyed to. Not to surprise. That is why both parts are in the same ability. Because they are both about winning initiative.
I thought it was quite a clever and elegant way to make an ability when I first read it. It works with advantage and sidesteps the issue of it not stacking by having a bonus for surprise.
The key is that the assassin must win initiative for assassinate to work.