There is no need to bleed the system, just turn the wheels all the way to left and right a few times, it will bleed itself. That's how you do that.
behind the power stearing pump. remove pump and you will see it
have you tried removing return line to bleed system ?
under neath the power stearing pump.
perhaps your power steering pump needing repair
check power steering pump or belt tension.
sounds like a bad power stearing pump
normally you keep turning the steering to full lock both sides and the pump should bleed itself.
the problem is that your saft on your power stearing pump broke and that afects your power brakes to
Have you checked the power steering pressure line?
To replace the power steering unit on a 1993 Pontiac Bonneville, start by disconnecting the negative battery cable and draining the power steering fluid. Remove the serpentine belt and disconnect the hoses from the power steering pump. Unbolt the pump from its mounting bracket and replace it with the new unit, ensuring all connections are secure. Finally, refill the power steering fluid, bleed the system to remove any air, and reconnect the battery.
Alternators do not generally over heat, and the do not contain fluid. Since it is red in color I would assume that you are spraying transmission fluid from a cooler line, or power stearing fluid. Power stearing is much more likely since the power stearing pump is also belt driven (just like the alternator), and commonly mounted very near the alternator.
That requires a special puller. It's called a power steering pulley PULLER. You can rent them at Advance Auto parts stores.