If you Jump it and it works, the issue is certainly the "Solenoid system" - That the solenoid is not doing what it is supposed to.
the Solenoid is very basic, it's an electrical switch, that when it is given a little bit of power to the little wires connected to it, it will "connect" the two big posts on it to each other, via an electric magnet, powered by the voltage to the little wires. So it allows a little electricity from the ECU to connect BIG wires together. When it works it just bridges the connections exactly like you did by hand.
Solenoid OFF (Big hole) --No Connection---(Big Hole)
Solenoid ON (Big Hole)---Connection--(Big Hole)
Are you sure you put the little red wire back on the correct post ? In my ECU the power to it is provided by a little red wire that goes to the live post too. If you forget that one, or put it on the wrong side, like I did, nothing works. Be sure the small red wire that was bolted to the post, if you have one, goes to the side that goes to the battery.
To test the solenoid with your tester, connect in the Omhs positsion (any resistance scale number is good 2K or 20K is fine) and verify it shows 1 to the left or some other indicator of Infinity, then put power on the little wires (12V) and you should hear it Click and the number shown in the multimeter should change to zero or close to it. (No resistance, Solenoid is connected. All good)
I'm hoping it's that you missed the red little wire, which is cheap and easy to fix.
Best of Luck!