I'm in an EU country and it's really rare to see a company do any background checks at all. Be honest, if/when you're asked about it of course - but just from personal anecdotal experience, it's rare that somebody even bothers to ask. (this might vary of course from country to country)
Until the idiots in charge change this (by enforcing the existing laws), you should take maximum advantage of its opportunities.
It's a rather ...unconventional suggestion, that clearly was not thought out. There is a high risk someone doing that ends up stateless, i.e., with no documentation or paperwork establishing citizenship in any country. The US may arrest such a person at the border - and then imprison them - and then try to deport them back to their country of origin. But what is your country of origin? It may mean a stay in immigration detention indefinitely. There's no guarantee authorities would believe a claim of US citizenship at that point. Citizens who've gotten in that position (usually cases of mistaken identity and/or misbehaviour by US customs) have been deported, and they don't always make it home. It is a potentially Kafkaesque nightmare and I would not advise intentionally putting yourself in that situation.