Yes, an intermittent duty solenoid will work, but will wear out in constant on duty. That would not be an issue in your testing-- it would certainly work that long.
OK, if jumping across the solenoid made everything work but it did not work with a new solenoid, you need to check the signal wire to the solenoid from the ignition switch.
Let me describe how this works: You have two large lugs on the solenoid. One always connected to the chassis battery. The other large lug gets power from the chassis battery only when one of the small terminals gets the "signal" from the ignition switch. Some solenoids have only one small terminal (signal terminal) and some have two small terminals. If two, the second is ground. If only one small terminal, the solenoid grounds through its mounting. If one terminal and not mounted on metal (i.e. ground) it will NOT work.
So, if properly mounted, you need to check for the signal (12 VDC) on the signal wire when ignition on, but 0 VDC when ignition off. That will tell you if you have a problem with the ignition switch or wiring from switch to solenoid.
You can also check by taking a small jumper wire (it will carry less than one amp) from the chassis battery lug to the small signal terminal. That will cause the solenoid to close and give power to those circuits downstream of the solenoid.