I do one better. I actually carry both the passport and the passport card when going to seedy countries

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I just walk with the passport card while having the passport itself in a stash spot in the hotel. I have the card physically strapped to my body or in a sock . This way a stolen bag means nothing. Even if they make you empty your pockets there's nothing of value on them.
The worst case scenario for you is that the hotel you're staying at burns down while you are on an excursion or you are personally incapacitated and unable to return to your hotel. In such situations You still need some documentation on your body that proves your citizenship when going to the US embassy in case sht hits the fan. They won't recognize the copy.
And yeah I totally agree with your burner phone concept . I remember taking a trip to Egypt shortly after the galaxy fold 2 came out. Because I was the only one in my whole city who had one, I swear that i loved that cool looking phone like a firstborn son, so it pained me to leave it in America. I took a cheap ass samsung a02.
So you can imagine my amazement when I went on to a tour group with a bunch of situationally unaware white people who were using the same $2,000 phone to take pictures in a slum surrounded by a crowd of people who dont earn 2,000 in a whole year.