Having a "photocopy" of your passport may shorten the time it takes to get a replacement by a day or so but it is worthless for transportation or even as a source of identification.
If you don't have your actual passport you will spend a couple of days wherever you are stuck before you fly home. It all depends on how long it takes the U.S. Consulate to issue a replacement or temporary passport. That is if there is even a U.S. Consulate or Embassy where you are.
There are risks to carrying your passport onshore and there are risks if you don't. The choice is yours.
If I'm going for any extended tour I take my passport with me.