Attached is a rough sketch of your control circuit connections. Your power wiring is connected correctly since you can manually get the motor to run manually, as long as the generated/wild leg stays on L2.
There is likely a problem with one of the control components. Sitting in a barn can allow oxides to build up on contact surfaces and cause them to be less conductive when inoperative, as you need contact arc to burn off the oxide buildup.
When testing refrain from testing to ground, this is mostly irrelevant as none of the controls or motor are referenced to ground. You need to do all testing from line to line, namely L1 and L3 in your case.
First thing to check is the control fuse. You should test from L3 to the top and bottom of the fuse. The top should be hot 240V, the bottom should read the same, if not the fuse is blown. This is likely due to you having low voltage at the start, then later dropping to no voltage. If its blown, test for a short circuit before replacing. A chattering contactor can also blow the fuse if it is sized closely to the coil draw.
The next things to test, again from the line voltage NOT GROUND.
ON/OFF switch, Its fed from the fuse L1 power leg, so one lead to L3 the other lead to both sides of the switch.
Pressure switch, also fed from L1 leg, so one lead to L3 the other lead to both sides of the switch. ON/OFF switch must be on.
Overload relay contacts, fed from L3, so one lead on L1 the other lead to both terminals of the overload contact.
Another issue that can cause chattering, to check last is the iron magnet assembly that is surrounding the coil. The contact faces can become rusty sitting in a damp location as they are raw steel lamination's. If this is the case the starter coil and magnet assembly will need to be disassembled to get the iron core out for cleaning. Clean the coil faces by lightly rubbing them against some very fine sand paper backed up with a flat piece of steel or glass. No excessive scrubbing or hand sanding. Blow the sanding dust dust off of all parts before reassembling the starter.
The coil should be good because it was trying to pull in while it was being supplied under voltage. just don't continue to let it chatter, it can overheat or continue to blow the fuse.
That should be enough to get you to the problem spot.
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