Are you following a repair guide? Removing the fan clutch involves a specialized wrench that clamps around the pulley nuts or in the pulley face to hold the pulley while the central nut is turned. This is one type, there are many other variants as well and you can typically rent them for free from a parts chain.
Had to buy the wrenches to remove the fan clutch
UPDATE
Pulled the water pump today and it looks like both gaskets (intake and outlet) failed. Bolts surrounding the gasket were covered in coolant and grime. Now just waiting on the two radiator hoses to finish (they are on order). Replaced serpentine and AC belts too. Looks like they were never changed.
Replace the thermostat and seal while you’re at it, unless it’s been recently done.
That comes with the pump. I had planned for that but the water pump NAPA sells (bad-ass cast iron with lifetime warranty) has the thermostat housing and thermostat built in so I got an extra.