Can’t find correct size outdoor spigot handle

Very odd, but here's a thought on how to troubleshoot. Shut off water pressure into your water heater. Test each one of your hot water taps. If any of them work, something is mixing cold water into your hot water pipes. Some one-handle faucet and shower fixtures can do this if their internal seals are worn out. Find it and fix it. If they all only work when water pressure is supplied to your water heater, then the problem is likely the water heater itself.

Are the other units on the same piping? If so, the mix may be caused by a recent change in their plumbing. Maybe you can engage them in the troubleshooting.

If you haven't stripped the screw head, they sell impact screw drivers for just such a situation. Like this one.

A tool like this allows you to strike the impact with a hammer while turning the screw. They use them for motorcycle cases with Phillips screws that otherwise would never come loose with hand pressure alone. It you could rent one for a few dollars it would save you some money.

Your next step is to drill out the screw and re-tap the faucet stem.

Good luck!