The Caps Lock light not changing after several presses doesn't necessarily mean any one thing. I had the same problem when I needed to reflow Alice's solder.
Compare the Amiga's boot flicker with that of an A1200. The monitor should lose sync momentarily, but it will be for about a tenth of a second. If the monitor just immediately resyncs without losing sync it may be the same problem as I had.
In my case because the joint was dodgy on Alice, it would occasionally boot up. Also leaving it on for a bit longer and then Ctrl-A-A would reboot it.
Do you have an oscilloscope? Check for activity on the SIMM sockets to make sure RAM is being accessed. If it's not, then the machine is probably waiting for one of the custom chips to respond correctly.
The machine should boot fine with just the PSU and the monitor connected, by the way. The keyboard is also useful but not necessary. You don't need the floppy connected.
Also, what is the voltage of the +5V line? It should be over 4.9V on an unloaded motherboard, though it may sink as low as 4.7V when everything is connected.
If you have an SMT rework station, I'd recommended a quick flux and reflow just to be sure. That's what fixed my A4000 (after years of trying to diagnose).
