I was under the impression that the ATMEGA chips produce a short heartbeat on startup regardless of whether a bootloader is installed.
Is this correct?
No, that's not correct. The heartbeat is from the bootloader code. An ATmega normally boots up with all the pins set as inputs.
I've never heard of a USBASP blowing up chips. Probably you managed to get an incorrect bootloader or fuses in them, somehow.