So the general reason for having a limited time to use an out-of-state (or foreign) licence appears to be jurisdictional.
Yes, just the same as a state gives you 10 days, or 30 days or whatever when you move to re-register your car and obtain license plates from your new state of residence.
I think the issue that most people have is not the requirement to obtain a U.K. license, but the way the system treats already licensed drivers as though they were learning from scratch and demands the full range of tests expected of a new driver. Couple that with the way the system grants full U.K. driving entitlement for the first year, and it just makes no sense.
The US won't exchange our licences so we don't exchange theirs.
As with every state, Nebraska requires a license within so many days of taking of up residency, but with the usual sort of clauses that existing out-of-state license holders may not be required to take the full range of tests -- Subject to the DMV's discretion and suitable driving record, etc. When I was there that was taken to include waiving the practical part of the test for an existing U.K. license backed by several years driving and a clean record.
The British system is too rigid: If you hold a license from an EU country or another country on the approved list you can just swap, even if you have a terrible driving record, but anyone from a country not on that list has to start at square one even if he's been driving safely for years.
I would be careful though, I'm not a legal expert, but it seems to me that all insurance documents include some declaration that you have a "valid licence" so if you declare that but don't (in the case of a 12 month + US licence) then they could invalidate the claim based on you misleading them, not any illegality.
Most insurers include all sorts of clauses to the effect that the policy will be invalidated if you don't comply with this or that, but my understanding is that regardless of those clauses they may not refuse a third-party claim anyway. I'm no legal expert either, but that's the situation as I've seen it set out.
I remember seeing something similar about a British learner (i.e. on a "provisional" license -- a learner's permit) and how it affected the insurance issue because he had told the insurer that he held a full license. The legal response was that while he could certainly be charged under the license violation ("driving otherwise than in accordance with a licence" if not accompanied, no L-plates, etc.), he could
not be charged with driving uninsured, since he did have a policy to drive the vehicle in question.
Frankly, as I see the situation an American driving on a U.S. license beyond the 12-month period is really only a committing what I would call a "paper offense." Yes, it's illegal, but it's not really hurting anybody. Thousands of British motorists commit similar "paper offenses" every day, by driving on an expired tax disk for a few days, by driving beyond the expiry of an MoT certificate for a week or two, by failing to notify a change of engine size, etc.