As far as I know, that official cross compiler supports all rpi models - so it might work with your current 1B. Something you could try while waiting for the new model? Up to you, of course.
Good point. But since building with an older cross-compiler that I successfully used to build an older version of Armory still results in the "Illegal instruction" stuff, that could indicate that it is assembler code in the crypto library that causes the failure. That the failure occurs at startup provides some evidence that the problem is not in said library, though.
I hope to get the new RPi today or tomorrow, but I will try a rebuild with the official cross-compiler if I get the time today.
EDIT:
The official cross compiler contains no installation instructions, no normal github releases, and no hints about how to use it. I could perhaps work it out myself in a day or two, but I think I will just wait for the newer RPi.
Various web pages let me to believe that it could very well be the compiler, apparently the RPi 1 contains an older ARM cpu that the normal cross compiler does not support.