LA on the Z80 is a good idea, if you've got enough channels.
It sounded at first like the lockup was related to the number of characters typed? But then it sounds like it might be strictly elapsed time related? So, it is sounding like basically "N" seconds after RESET the system locks up? I'm still thinking DRAM refresh, and the early K10 (ROM 1.9E) has discrete components for the DRAM (not the custom LSI chip) so you have a better chance of diagnosing and fixing it, if it is refresh.
It looks like this version of the ROM is continually checking for bootable media - although it's possible that whatever is wrong with the HDD results in a hang - so it might be interesting to leave the system "at the ROM prompt" for long enough to reach lockup, then insert the floppy and see if it boots without pressing RESET. Of course, if refresh is the problem you may be able to run out of ROM for an indefinite period of time (depending on RAM access patterns - if the RAM (e.g. stack) is continually being accessed then it won't "fade" even if there is no refresh). But as soon as you alter the code path (e.g. press a key) then you likely access RAM (via RET) that has faded and you crash.
Also, the ROM does have some snippets of code that it copies into high RAM, I think (other versions do, at least), in which case some event may cause that code to be executed after it has faded. Of course, once CP/M is loaded then nearly everything of import is in high RAM, but the console input routine (for example) is in ROM and may be able to survive while spinning (but as soon as you return, you crash).