I got really excited about what you said here so I wanted to verify. While you are correct that the PHB doesn't say being surprised ends from your turn happening, the official Sage Advice Compendium does:Then initiative is described and nothing in that description says that your turn coming up in initiative order ends surprise.
Unfortunately, being first in combat does end your surprise and an Assassin Rogue cannot benefit from it.Sage Advice Compendium pg 9 said:If anyone is surprised, no actions are taken yet. First, initiative is rolled as normal. Then, the first round of combat starts, and the unsurprised combatants act in initiative order. A surprised creature can’t move or take an action or a reaction until its first turn ends (remember that being unable to take an action also means you can’t take a bonus action). In effect, a surprised creature skips its first turn in a fight. Once that turn ends, the creature is no longer surprised.