In all cases I highly recommend that a newly minted citizen inform all parties of interest of the person's adjustment to citizenship. I don't care if it is required at the moment or not. The reason I believe this is because one never knows what sort of decisions will be made in the future about yourself by another party based on your immigration status as it is recorded in their system. For all we know at some point in the future LPR's could be denied certain currently existing privileges or an employer may decide that in a highly charged environment of war that the employer only wants citizens in their employ. There could also be grants, awards, etc which are only available to citizens, but you didn't notify the interested parties of your citizenship, so when they are notifying about the awards, grants, etc, they neglect to include you. There are many possible scenarios and all of them can be avoided by just informing all interested parties now that you are in fact a citizen.
I for one, upon becoming a citizen, informed my employer (i-9), informed Social Security, informed the FAA (i have a pilot's permit), and notified a few others i deemed relative.
I still think you are thinking like a visitor not like a citizen. The employer shouldn't be using the I-9 for any purpose than to verify eligibility to work--and so it really shouldn't matter at all.
Yes, in practice, it may not always work out like that. But to the extent that the unpredictable intangibles you mention come into play, I think updating the I-9 will hurt, not help.
The reason is that updating an I-9 draws unnecessary attention to the fact that you are not originally from this country. I can pretty much guarantee you that, in the mind of the HR person, they think of updating I-9 forms as being something that foreigners do--it's not something that citizens do. Citizens fill in the I-9 once when hired and then they're done with it. It puts you--not legally but in the mind of the HR person--back in the same category with the folks who need repeated sponsorship for H-1B's or renewals of EAD's, etc.
So right at the moment when you have gotten your citizenship and want to exercise the full privileges of US citizenship, you've unnecessarily reminded HR that you are a foreigner 5+ years after the last time the issue came up. If you want your career to grow into roles that require US citizenship, that's absolutely the last thing you should do.
Of course,
legally you are no longer a foreigner, you are a citizen. But I guarantee that in the minds of the HR person, you've now completely unnecessarily tagged yourself as a foreigner all over again, and have set your career back 5+ years. Very foolish thing to do.
New citizens often want to celebrate their citizenship and there is definitely a temporary high or euphoria to it that they want to share with everyone around them. And that's great when it comes to family and friends. But the HR person is, in general, not your family or friend. Updating the I-9 with the HR person will, far from causing them to think of you as a citizen, strongly reinforce the (now incorrect) notion in their mind that you are a foreigner.
Updating government databases like the SSN card and the private pilot records is, obviously, a different matter and not likely to cause a problem. It's a different matter when you are updating records with people you work with every day and whose subtle perceptions of you may affect your job.
Indeed, for the same reason, I'd strongly recommend that when you get your GC (don't wait until you are a citizen) you update your SSN records to get an unrestricted card. That way if you are hired as a GC holder, you show the SSN/DL (not the GC) when filling out the I-9 form. You'd still need to indicate yourself as a GC holder on the I-9 form itself, but they will definitely think of you as more "American" and less "foreign" if you show ID typical of Americans.
