Why will this vehicle not start?

Alright. Vehicle is a 93 Ford Explorer. Radio, lights, etc. All working fine when key is to ON. When the car is to start, all we hear is a click. We've pushed to start and that's worked half the time. Cleaned up the battery terminals.

Battery gets 12.6V standing and 11V when trying to start. Had starter checked twice, worked. Replaced fender-mounted solenoid. Tried bridging the solenoid, same click. Got 11.2V on the starter solenoid terminal ('always hot') when not starting.

Headlights don't noticeably dim. Couldn't jump-start, either.
Thanks
Added (1). That could be. I wonder why we still get 11.2V, though. When starting, the current between the "always hot" and only-cranking terminals zeroes out. (I'm thinking that the current was flowing properly between those two. Either that, or starting actually turned it off.) Didn't test the third.

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