The diagnostic rom will run on a machine with no RAM in it. It's designed that way to help repairers. There are other diagnostic ROMs and cartridges that do different things, and that particular ROM is designed for it's own hardware, but I've tried it with just an external ROM card I made up and it works fine for the critical tests. I have some schematics too for a home-made external ROM adapter if you have access to a CNC machine or some single-sided circuit board making equipment - I can share it if it's of use.
But plugging the ROM straight into the socket on the PCB should work too, though I haven't tried that with an Issue 2. It should ignore the select on 27 though, and will run on just the ROMCS and the RD lines from the CPU. When I said "Border" I meant that if the RAM is completely broken, then there's no way for the ROM to display the results, so it times and flashes the border different colours, which the ULA can handle even if there's to video RAM. By adjusting the timing, it creates a result like the "Loading Bars" that you see on a spectrum when loading, but it times them with an interrupt so that they are static, and then shows which are working and which are not by either the number, or I think that particular ROM shows some as RED and some as GREEN and the top bar represents D0 and they count down to D7 or something like that.
I do recall you saying you had tested the RAM, but if they are installed and you lose the -5v and 12v lines, then it can actually damage them, and it can do so very very quickly. So if you lose TR4, it's always an assumption that even if the RAM were tested good, that they might have been damaged in the process. Since you have an AppleII to test them, I'd definitely do that.
The ROM, BTW, identifies RAM faults, but the fault may be aside from the memory chip - It could also be a broken trace on the PCB, or another chip ( eg, the 32K ram chips ) interfering.
The Upper 32K memory chips aren't obtainable any longer, but don't worry - they are just 4164 RAM chips with pre-existing faults in half of the chip. ( hence why they sold them as 32K chips ). You can replace a 32K ram chip with a 64K ram chip without modification and you can mix and match them also.
You can even go as far as replacing 4116's with 4164's as long as you fix up the power rails and move the power lines around, which is entirely doable on the Spectrum.
The 12v line from the onboard regulator only powers the lower RAM and the Video circuits, so it's not critical to any other chips.
Removing the upper RAM will eliminate it from causing problems if there are any faults in it - Faults could be in either the upper RAM chips themselves, or in the 4 address logic decode chips ( 4 of the group of 6 ) that control it's access and multiplexing. If for any reason there's a fault in the upper RAM, it may interfere with the data bus and cause the CPU to not execute correctly. Since you have a rather complex problem to fix, removing the upper ram will at least eliminate one potential source of issues and testing/checking/repairing the upper ram as a final step is easy. Unlike the lower 16K RAM chips, the upper 32K is directly on the CPU bus, like the ROM. I had one board I fixed, and never quite managed to find a niggly issue that caused problems, and traced it back to a bad upper memory chip that was simply interfering with the bus at times. If the RAM and selecting / multiplexing logic is all OK, then there's no problem leaving it there, but it's difficult to know for sure until you can get the machine at least booting to logo, and it's easy to eliminate as a potential source of issues ( and you may have multiple faults ).
The final path to getting a Spectrum working, once you have the CPU and ROM working OK, is sometimes just tracking down every trace on the board and testing for continuity -The Lower RAM goes to the ULA and to the resistors that separate the ULA from the Z80 bus. Also, those resistors have been known to get broken before so also test those resistors.
You seem to have a functional video circuit, so the rest of the board should be easy enough to fix once you are confident that the CPU and ROM are working as intended.
BTW, the 74LS157 Multiplexers are also something that frequently breaks, as a fault in the DRAM can introduce 12v or -5v to other lines such as the MUX'ed input address lines, and these go straight back to the 74LS157's as well as the ULA.
Regards
David