The priority flow valve Jim describes may explain why your power steering and brakes could be acting normal even if the pump has been damaged.
Suppose the pump, when operating properly, produces 10gpm at 2000rpm, and that the priority flow valve is set to send the first 2gpm to the power steering and brakes. If the pump has been run dry or nearly dry, the first damage is usally between the ends of the gears and the ends of the pump housing since that is where the metal surfaces slide on each other. The lack of lubrication causes the metal surfaces to touch and score, destroying the seal between them and allowing fluid to escape around the ends. When the fluid gets low, it also heats up because there is not enough volume to properly remove heat from the system. This reduces the viscosity and further encourages scoring of the surfaces. Some pumps have replaceable wear plates on the ends, often of soft metal like brass, that suffer most of the damage and can be replaced.
In any case, if the pump has been damaged, the leakeage will greatly reduce the pump output, particularly at working pressures. The pump may still produce, say, 8gpm if there is no load. But if the pump is forced to work against even a small restriction, so that the pressure raises to even 100 or 200psi, the flow will drop dramatically.
That is where the priority flow valve comes in. The typical spool type valve forces fluid to the priority circuit (PS and brakes) by cutting off all flow to the excess circuit (three point hitch) until the desired volume is going through the priority circuit. This is the resting state of the valve; that is, when you first start up the valve spool rests under spring pressure in a position to send all fluid to the PS/brakes. Only when the flow to the priority circuit reachs the design volume does the pressure differential in the valve cause the spool to shift so that some fluid is sent to the excess circuit. Your pump could well be producing just enough volume to satisfy the priority flow setting, but leaving no more flow for the excess circuit. If that is true, the power steering would still work, but the three point hitch would not operate since it would not be receiving any fluid.
I doubt that the problem is air in the pressure line from the pump to the TPH. The pump must be getting fluid from the reservoir since the PS is working. If the pump is sending fluid to the TPH any air in the lines will just compress until its pressure equals the fluid pressure, and the working device (TPH ram) will still be under pressure. The air may make the device move jerkily, but it will move, and after a few cycles most air will be removed from the system. This is different from air in a car's brake lines, since the volume from the master cylinder is limited and will not compress the air very much, while the tractor pump will continue to force fluid into the line until the air pressure equals pump pressure.
The first thing to do is to check the pressure at the pump, ahead of the priority valve, with the wheels turned to lock. This will cause the priority flow control valve to close off the excess port (if it is not already doing so) in an attempt to keep the priority flow moving through the steering circuit. This should do two things: 1. give you the maximum pump pressure, and 2. cause the system relief valve to open, causing the familiar buzzing sound. I suspect you will find that the pressure is below specifications and that the relief valve does not open since the pump is not building up enough pressure to reach the valve setting.
Good luck. And keep us posted.
Transporting the tractor is probably the hardest part. Maybe a visit with their service manager could give you some good advice.