I seem to be reading surprise differently to other people. The way I read the rules, if a target is surprised it lasts for the whole of the first round, it doesn't end when they have their turn.
Nope. Surprised creatures roll initative just like anyone else at the start of the encounter.
When their turn comes up on round 1, they simply cannot move or take any actions (they still get a turn on round one; they just cant do anything with it). After that turn ends, they can then take reactions.
Its not just semantics either; they might be hit by a spell on Round 1 (before their turn of inaction) that allows a saving throw at the end of their turn, or they may be subject to some other effect that happens on their turn. The fact they get a turn (they just cant do anything with it) is important for the working of a lot of rules.
Consider also a surprised Wizard. With a high enough Initiate check, they might be able to cast
Shield during round 1 against an incoming attack that round.
To trigger Assassinate, an Assassin needs to Surprise his target (catch them unawares)
AND roll a higher initiative score than his target (so he acts before the targets first turn on round one) because as soon as his potential target has his turn of doing nothing, he's recovered enough to be immune to Assassinate (and can take reactions).
Its why feats like Alert, a 3 level dip into Gloomstalker, weapons of Alertness, the UA Battlemaster manoeuvre 'Ambush' and other initiative boosting tricks are invaluable to an Assassin.