Could have multiple issues and "new" parts are frequently junk these days.
The solenoid is probably bad either right out of the box, or trying to start it with issue(s) could have welded the contact together again.
Battery needs to be fully charged, starter good, cables good, or very easy to kill a "new" Cheapnese solenoid.
The key on until battery went dead can kill a module.
Having voltage @ both terminals of coil doesn't necessarily mean it is bad.
Did you check for spark?
If the wires on solenoid are reversed I believe will cause the starter to run/engage with key in on position. But it has been many years since I've had a vehicle with that style system so maybe incorrect.
Honestly throwing a bunch of parts @ it may just cause frustration and introduce additional issues. The aftermarket parts of today are a big roll of the dice.