You can drive across the border to Mexico without even stopping. No one checks. Just don’t have food or drinks that’s all they seem to stop people for

I don't think you actually need a passport to seek asylum in another country, it just makes the process smoother. If course you'd need one to use an airport, so you'd need to pick a country you can drive/walk to if you don't have connections.

Of course that's for refugee status, which i personally say most Americans should be entitled to even before the new nazi party took over but others countries might not agree yet.

You may wish to take the passport with you, maybe eventually things will change in US, or you will need to reference your citizenship. For instance UK home office refused to change my name without having it changed in my other passport first. Even ongoing travel restrictions to the country involved did not make them vaiver.

So from that perspective it might be useful for you to have the US passport. I have now accepted that every 10 years I gotta pay my birth country tax... In cash... And spend at least 1 day trip to the embassy to get the process started again. Even if I completely disagree with their politics.