The PHB Pg. 189 which describes surprise does not support the idea that your turn coming up in initiative ends being surprised. The order of operations on that page is that the DM determines surprise by comparing any stealth checks to passive perception. Then positions are determined, then initiative is rolled by everyone (including the surprised creature(s)). Then characters take their turns in initiative order. If a creature is surprised, 'You cannot move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends'. A member of a group can be surprised even if others aren't.
Then initiative is described and nothing in that description says that your turn coming up in initiative order ends surprise.
So, for the sake of an example, Our assassin; We look at the Assassinate class feature. Pg. 97 of the PHB. 'You have advantage on attack rolls against any creature that hasn't taken a turn in the combat yet. In addition, any hit you score against a creature that is surprised is a critical hit.' Our surprised guard rolled well on initiative but because he is surprised does not move or take actions on his turn. True, that doesn't mean his turn in initiative order didn't happen, he's still taken a turn, but this rogue, a hidden attacker (as defined on pg. 198) gets advantage against the guard because this rogue is a hidden attacker. The guard is also still surprised because nothing in play, or occurring on his turn has ended surprise. This assassin rolling 3 on initiative goes last in order, and attacks the still surprised target from hiding with advantage. He hits, and his hit is a critical hit because the target is still surprised.
This is a good set of questions, and a good ruling, but it's not a necessary one. We need to take a look at what Surprise does in the rules. Surprise is not a condition (which would make things easier) and, to compound this, it doesn't have clear exit criteria. As such, Surprise sits in a weird little place in the rules -- it's not really one thing or the other; it's its own thing. So, then, let's look at what it actually
does.
Surprise really only does two things. It (1) prevents a Surprised creature from taking actions on its first turn in initiative; and (b) prevents a Surprised creature from being able to use reactions until after it has completed its first turn in initiative. So, it really only affects what a creature does on it's first turn with the exception of the reaction prohibition prior to it's first turn. After it's first turn, Surprise does nothing at all. A rule not doing anything can be considered to not be applying. This is the basis of saying that Surprise ends at the end of the Surprised creature's first turn -- Surprise stops doing anything at that point, so it would appear to have ended.
And, absent the Assassin subclass, this really wouldn't be any kind of issue. However, the designers decided to rest a subclass' signature ability on an ill-defined state in the rules. And, this is where your ruling comes in that Surprise lasts until the end of the first round. This certainly isn't contradicted by the rules (Sage Advice notwithstanding), so it's a fine thing to rule. And, it aids the Assassin, in that it removes one of the three existing gates on its signature ability (the first being gaining surprise, the second being winning initiative, and the third being hitting the target successfully). I think that's dandy, and have toyed with implementing that ruling myself (currently, the Assassin/Gloomstalker PC in my game has a ridiculous bonus to initiative and advantage on initiative checks, so it's not a pressing need). It doesn't hurt anything to rule so. The only thing it does is add a touch more artificialness (made up word alert) to Surprise, which is already suffering over unclear exit criteria, limited effects, and being poorly named. Still, it's not like adding that ruling makes it worse -- arguably it makes it a bit clearer to use with regards to Assassinate.
Finally, if you have an Assassin that can roll a 3 on initiative -- perhaps that's the exact outcome that player was courting with that particular build?