I don't know that we have the data on Omicron making people sick enough to realize it after they've been boosted. We definitely know without a booster it happens. We also definitely know boosted people who test for other reasons are coming up positive with Omicron, and reporting no symptoms. These are in the news a lot.
Either way, you're probably right that the chances of catching Omicron twice in a short period are way less than your chances for the first one. Probably true no matter your vaccination doses, but even more so with a full boosted course.
A year later, harder to tell. But, you're not going to be a year later for a year to worry about it.
Which just gets back to, if you're going to make decisions for after recovery based on if it was COVID or not, you should get tested. If it's not going to matter how you make decisions after recovery, then it doesn't matter. Either way, as long as people act like it's COVID while they're actually sick, that's fine.
Nothing wrong with both approaches.