If it just breaks the game for you if someone takes a shot outside of a rolled initiative order, then you can always burn the assassin's reaction as a held attack. Assassin sneaks up on target, is "ready" to attack. You do your initiative thing, then the Assassin goes first with his reaction to take the attack he's "readied," then proceed in initiative order. This means that the assassin will get another attack (or two with a bonus action) in the round, but his reaction is burned so if the target survives the alpha strike and wins initiative he can move away without provoking an attack of opportunity.
The bottom line is that the assassin, if he gets surprise, should get a guaranteed crit if his attack hits. Otherwise, you're screwing the assassin big-time. The assassinate feature doesn't say "if you win initiative," and it is a profoundly lame ruling to declare that the signature feature of the assassin is dependent on the initiative roll.
The rules are guidelines. That's how the rules work. It's up to you, as the DM, to make sure that those guidelines don't get in the way of the game.