No crank starter clicks but doesn't crank, benched tested starter and is fine, motor spins, still no crank

Take the starter off and go through it again, with a no crank condition its pretty simple in that there is only a few components that are going to prevent cranking over, starter and starter solenoid, battery, and ignition switch. If its clicking thats most likely the starter solenoid clicking but the starter motor isn't spinning. Bench testing a starter only tells you that it will spin without any load on it, but on the engine it might not have enough power to spin. Check all cables, clean all connections, check voltage at the starter terminals and compare to the battery voltage, if there is a big difference you have a bad connection somewhere causing voltage drop.

If all else fails get a new starter since its a new battery and go from there. Having it out on the bench without a load might have got it unstuck and working again, still sounds like a failing starter. Try giving the starter a few taps with a hammer on the side and try starting again to see if its stuck.

How is the battery and connection?

low battery, battery cables not making secure connection, battery terminals oxidized or needing cleaning/brushing .... failed starter .... tap the starter with a wrench, it can revive it :)

the problem is that it is making contact in the solenoid housing before the pinion is engaged in the ring gear
have that measurement checked again as the pinion has to be fully extended before the contacts are closed to spin the starter armature

If you have tested the starter, and it is installed correctly, and it spins, my guess would be missing teeth on the flywheel.
A Ford starter needs 11 volts to engage so you could check to make sure you have that at the starter cable post.
You could also turn the crank a few degrees to see if the flywheel is warped or damaged.

If you have power to the starter motor solenoid, yet it will not start, then replace the solenoid.
If the solenoid is clicking when you turn the key, but if it still doesn't crank, then change the starter motor.
The solenoid is on the top of the starter motor.

You have one of two problems: Either the starter drive on your starter is defective and is not engaging the flywheel to crank the engine, or the flywheel has some broken or damaged teeth that are preventing the starter from engaging.
Starters come in a variety of designs. On some, the solenoid is mounted on top of the starter. When you turn the key, the solenoid routes current to the starter motor and at the same time pulls a lever that slides the drive gear mechanism out so it will engage the flywheel and crank the engine. If the solenoid is weak or damaged, it may not be strong enough to overcome the spring tension that retracts the drive gear. So the starter spins but doesn't crank the engine.
On other starters, the solenoid is mounted remotely. When the starter motor starts to spin, it ratchets out so the drive gear will engage the flywheel and crank the engine. If the drive mechanism is damaged or hung up, the motor may spin but not crank the engine.
Regardless of what type of starter you have, it will have to come out for further inspection. The drive gear (which is sometimes referred to as a "Bendix drive") should move out when the starter starts to spin. The drive gear usually has a one-way clutch that is supposed to protect the starter against damage if someone keeps cranking the engine once it starts. The gear should turn one way but not the other. If the gear is locked up or turns freely either way, the drive is bad and needs to be replaced. If the drive can't be replaced separately, you'll have to replace the entire starter.
Starter Testing If the drive seems okay, the starter should be "bench tested" using jumper cables or special equipment designed for this purpose.
CAUTION: Be careful because a starter develops a lot of torque. It should be held down with a strap or clamped in a vice (be careful not to crush or deform the housing!) before voltage is applied.
A simple no-load bench test can be performed with a battery and a pair of jumper cables to see if a starter motor will spin. But this test alone won't tell you if the starter is good or bad because a weak starter that lacks sufficient power to crank an engine at the proper speed (usually a minimum of 250 to 500 rpm) may still spin up to several thousand rpm when voltage is applied with no load.
A better method of determining a starter's condition is to have it tested on equipment that measures the starter's "amp draw." A good starter should normally draw a current of 60 to 150 amps, depending on the size or power rating of the starter. Some "high torque" GM starters may draw up to 250 amps, so refer to the OEM specifications to make sure the amp draw is within the acceptable range.
If the starter does not spin freely, or draws an unusually high or low number of amps, it is defective and replacement is required.
An unusually high current draw and low free turning speed typically indicate a shorted armature, grounded armature or field coils, or excessive friction within the starter itself (dirty, worn or binding bearings or bushings, a bent armature shaft or contact between the armature and field coils). The magnets in permanent magnet starters can sometimes break or separate from the housing and drag against the armature.
A starter that does not turn and draws a high current may have a ground in the terminal or field coils, or a frozen armature.
Failure to spin and zero current draw indicates an open field circuit, open armature coils, defective brushes or a defective solenoid.
Low free turning speed combined with a low current draw indicates high internal resistance (bad connections, bad brushes, open field coils or armature windings).

The starter sprocket wheel is thrusted into engaging the flywheel by a big solenoid. That solenoid is integral part of the starter motor. If the starter turn itself but it doesn't turns the engine then either it was incorrectly installed or the thrust solenoid is defective.
Simple test: take out the starter, attach it to a bench, connect to it thew power cables from a battery (the minus one to the body, the plus one to the big connector bolt) then short the big connector bolt to the small one with a screwdriver - the solenoid should activate, the sprocket should be thrusted forward and the motor should start spinning it.
If that doesn't happen then the starter you just bought is defective, replace it. If that happens the starter was incorrectly installed on the engine. Reinstall it correctly.

maybe starter relay?

First make sure you can turn engine over with a socket and that it is not seized. Second if engine checks out fine does the starter just click with the ignition key? If it does perform what is called a voltage drop test on the starter cables. Connect a volt meter on each end of the cable you are testing you should read no more that a half a volt drop when loading the circuit. If cables check out fine pull the starter and have it bench tested at an auto parts store. But be carefull bench testing can show a starter is ok with out putting a load on it, if it still shows good on the bench you still may need to replace it. If you don't get a click with the key check for 12 volts at the starter connector unplugged, if no trace starter wire noting color code back through the starter safety switch back to the ignition switch.