I do this same thing (using the initiative of the first character to act) is situations like opening doors or other transitions into combat. i.e. the first character that it is important to move to combat rounds is the first one to act and the round begins with their initiative count.
Newbie DM here running 5e. Last night my players were moving stealthily through a dungeon and the two characters in the lead both perceived the dim torchlight of an approaching enemy from around a corner. This wasn’t visible to the other party members (who don’t have darkvision), and the two characters who were aware of the imminent threat neglected to inform the other characters.
Sort of does. But since they don't get to act during the surprise round anyway and since we do cyclical initiative, it's just for placing people in some other than random order anyway.That would seem to penalize participants who roll initiative higher than the first character to act. That seems odd. Or maybe I am misunderstanding your description.