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.
My understanding is the main reason there is an initiative roll in the surprise round is that some targets may be surprised whilst others may not be.
"If you’re surprised, you can’t 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 the other members aren’t."
Seems pretty clear to me.

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