OK, the first thing to do is to double-check what you have removed / refitted as part of your previous fault-finding with the ignition switch. While not 100% guaranteed it is very likely you are now dealing with something you have introduced given you had no issues with starting before this exercise?
You have a relatively healthy system to have the fuel pump prime and having absolute zip from the starter is pretty straight forward to isolate.
You mention having power at the Black / Red wire so I'm guessing you measured this on the relay base for the Start relay when the starter button is pressed?
What you need to check is continuity from the relay socket (the opposite small socket, the Tan / Light Green wire - you are dealing with the outer two of the three small ones) to the battery ground. This should be zero ohms when the clutch is pulled in. The diode will make it tricky to measure the neutral switch from this point as you will need to have the meter on Diode Test rather than Ohms (the -|<- symbol) with the red multimeter lead on the relay base and the black lead on the battery ground. On my Uly that measures at 754 ohms when in neutral and open when in gear.
You mentioned earlier about the neutral light flashing for a second when you started it so that sounds like it might have a dodgy connection or switch? It has a push on connector and the switch is located behind the front pulley cover at about 1 o'clock just behind the pulley. This is really easy to test. Pull the connector off and with the ignition ON grounding this wire should turn the dash light on.
With the meter on resistance measure the connection point on the neutral switch to ground and it should be very close to zero ohms in neutral and open circuit in gear.
It seems really unlikely that you actually have a starter solenoid issue but you can verify that by momentarily bridging the the two large sockets on the START relay base. Now, be careful about doing this. When you do this the solenoid should pick up and the starter begin to crank so I am just taking about a touch so you can hear the loud clack from the solenoid.
The one obvious place where you could have crippled the starter system is via swapping the relays. If you were really unlucky you
might have swapped a bad relay into the start relay position in the attempts to chase down the ignition switch issue. You might want to grab a piece of chalk and mark each relay with a 1,2,3,4 to establish their position now and cycle each one through the Start relay position.
I have added a photo of the relays for my Uly and they are going to be a different overall layout but the contacts I am referring to for the power and ground are marked here.