Ability checks, like initiative, are only needed to resolve uncertainty. In a situation where only one character is trying to start combat, there's no uncertainty regarding who starts combat for an ability check to resolve. Accordingly, in the situation described I would have the initiator at the top of the initiative order, and have everyone else roll to determine the rest of the order. (In a different situation, where multiple characters wanted to react to the same IC stimuli by starting combat, I'd roll as normal, so there's no advantage of being the first to shout out that you attack.)
My approach comes at a cost: in situations with one initiator it becomes impossible for someone else to act faster. However, I'd point out that all methods for transitioning to initiative with one initiator come at a cost: if you want to keep the ability to pre-empt the initiator, you have to either: (1) lock the initiator into their declared action; (2) allow the pre-emptor to react to a declared action that may never actually happen; or (3) let the initiator start--but not resolve--their action prior to transitioning to initiative.
From a RAW standpoint, I think (2) is arguably correct, but it comes with the additional complication of what information you give to the pre-emptor: do you let them act on the knowledge of the initiator's declared action, even though the initiator may choose not to take the declared action when it finally gets around to be their turn? If so, you introduce weird causality issues, but if not, going first in initiative may be a disadvantage if you don't know enough about what is coming to make going first useful. (In the most extreme case, if you have to skip your turn because you don't want to initiate combat and IC your character doesn't know that the initiator is going to, the end result of winning initiative is the same as going last in initiative.)
I think the cost of not allowing pre-emption when there is only one initiator is the least disruptive of the available options, but that's strictly personal preference.