I agree with removing the solenoid you’ve got there as well as all of the wiring from the key ignition to the starter.
I once had the exact some issues you have. My fix was simple, easy to do & solved the problem long term. Never surfaced again…never. This was on my FJ 62…but the principal is the same. No more click, click, click.
Go to auto parts store & buy a $15 four or five pin RELAY. You only use 4 pins but either a 4 or 5 pin will work. You will make 4 wire connections.
1) one from the ignition wire where you put in the key to start
2) one to a good ground
3) one directly to your battery positive
4) one directly to the starter solenoid
That’s it. I’m pretty sure I’ve described it correctly…it’s been 9 years now. If you go this way, do some google searches. It’s a very simple fix…you will be getting power directly from the battery to the solenoid and with fresh wires. Here’s a couple of pic.
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