Arial Black
Adventurer
If you allow initiave to determine when surprise ends, then it isn't literally hearing the "Bang" that allows the target to be unsurprised if he beats the assassin's initiative. Narratively it's perfectly acceptable to suggest the assassin gave himself away during the process of attacking. So yes, the the assassin was stealthy enough to surprise the target, but he was too slow or clumsy on the actual attack. The target notices the attack in progress and can react.
The bolded part is a big problem.
What determines (in rules terms) whether the assassin gives himself away or not is the result of a contested ability check, usually Stealth/Perception.
Your assertion that 'the assassin gave himself away' assumes that he does give himself away no matter how skilful, and that the determinant of whether he gives himself away is not his great stealth nor the victim's poor perception, but who rolled the highest initiative! That if the assassin rolled a high initiative then he's stealthy and if he rolled a lower initiative then he's not stealthy?
Initiative has no bearing on stealth, conceptually OR in the rules.