Great news !
So this will let me think about what's wrong.
So the ASZMIC is working on position "E" - right ?
The ZX80 should be working on position 4 and 5, the ZX81 on position 6.
All positions have the WR line set to high - so wouldn't possible to write anything to the EEPROM.
Normally can not be destroyed because you need a kind of password to write data to it but there are some commands in ROM which make the system to hang while trying to write to ROM (which is normally not possible, some bugs in the firmware from Sinclair). So PRINT is a candidate. So better to leave the EEPROM write protected.
Hmm now I have to thing.
About the oscilloscope - maybe you can borrow one from someone you know for a few days if necessary.
10 MHz is not the best but better than no scope.
For finding problems on the board it would be sufficient as most signals are in the 1 MHz range or below.
If you anyway are interested in an oscilloscope you should try to get one via Ebay.
I got mine 60 MHz dual scope (analog - would be the best) for about 100 EUR, is a Tektronix.
So if you get the video signal in good condition from ASZMIC - the ROM should work at all, same for RAM and the video part.