Yes, other countries will keep working with the US, but with a different attitude. They will be much more weary of doing long term things with the US unless they will get enough benefit in the first N years (where N is the remaining term of the current president) for it to be a net win even if the thing stops abruptly after that.

Trump 2 is a breakdown of the separation of powers and checks and balances that were supposed to give the US continuity. With that no longer functioning any deal you make with the US is really a deal with the current President which future Presidents may or may not honor.

It doesn't matter how reasonable the next President is because just electing a good President doesn't fix the underlying system. If other countries don't see that fixed so that another Trump cannot happen they will not consider the US to be reliable for long term things.