'65 Formula S Power Brake Booster Problem

I think the giveaway is the hard pedal, lack of pedal travel and, lack of Idle change upon pedal application.
Lets start with the lack of idle response. I'm so glad you mentioned this.
The reason for the engine response is because:
NORMALLY, when you apply the pedal, there is about a half an inch of pedal travel, before anything happens. During this time, the pushrod from the pedal, goes out to the atmospheric valve inside the booster and tells it that you would like it assist you. What the atmospheric valve then does is open a port to atmosphere which is at about 3> 4 times the pressure as is on the vacuum side. So in it comes and as it does so, the vacuum side sends a signal back thru the checkvalve, and if it doesn't respond fast enough, it momentarily upsets the intake. But as soon as you take your foot off the pedal, the atmospheric valve closes, and then, the intake manifold has to evacuate that chamber again to pull the diaphragm back into neutral, which simultaneously pulls the power piston that is inside the M/C, back to it's assigned parking spot. This is what the engine is really responding to.
So this is how it is supposed to work.

Now lets talk about the M/C.
This gizmo is nothing more than a hydraulic cylinder and works more or less exactly like a bottle jack. Except;
Inside the reservoir, down in the bottom, at the firewall end, is a tiny hole, called the compensating port. Every time you apply the pedal, a tiny amount of pad/shoe material gets rubbed off. If there was no compensating port, then, as the pads wear off, the pedal would sink ever lower and eventually would end up on the floor. The Compensating Port compensates for that by allowing fresh fluid into the pressure chamber, at the beginning of every stroke, which is why you get that half inch of nothing when you press the pedal.
Now, if it should happen that the fluid that went out to the wheel cylinders did not return, then the next time you apply the pedal, as soon as the power-piston closes the C-Port, the pedal would be instantly hard. So then, you would still get the half-inch of pedal-travel, but that's all.
Furthermore, if the travel is NOT enough to open the atmospheric valve, then you would NOT get assist, nor the subsequent aberration in the idle-speed.
But if the power-piston simultaneously could not be pulled back to it's parking spot, by the spring-loaded diaphragm inside the booster, then you wouldn't even get that 1/2" pedal travel. I suppose that, this could happen if the slave cylinders were ALL siezed, or all the hoses were internally blocked, or some combination of that, but this is highly unlikely. But if they did seize in the applied position, then the shoes would have smoke-signalled you.
But you still gotta jack the car up and check that the wheels turn freely. If they do, well that's a good sign.

So now, lets begin the diagnostic.
1) The first test for me, is to test the Booster. See note-1
2) The next test would be to look inside the M/C and make sure the C-port is open and unobstructed by the power piston. If you cannot see the shiny piston, it's probably ok, but I would stick a thin wire into the hole. The floor of the M/C is only .25 or less thick so if the wire goes in a half an inch then yur all set.
3) The Next test is to see if the M/C is moving fluid, see note-3

Note-1
The booster test
To do this, we need to normalize the booster, by getting all the vacuum out of it. I do this by pumping the pedal several times with the engine off.
If the pedal will not budge, see Note-2.
So now, lets say the booster is normalized, the engine is off, and the pedal is operating normally; here's how you test the booster. You step on the pedal with modest amount of force, hold it there, then start the engine. As soon as the engine fires up, manifold vacuum will enter the booster, and because you have opened the atmospheric valve with your pressed pedal, the pedal will fall, as atmospheric pressure overcomes the vacuum ..... if it is all working correctly.
If the pedal does not fall; one of three three things could be wrong;
1) There is no vacuum signal, or
2) the Pushrod to the M/C cannot physically move, or
3) the unit is defective. If you hear ANY hissing under the dash, the diaphragm is ruptured, or the valve is broke.

Note-2
If your pedal will not budge, make sure it is fully retracted, and parked at the top of it's stroke, and is not seized on the pivot shaft. If it's ok, yet will not budge, then See note-3

Note-3
Perhaps there is already pressure trapped in the lines.
to see if that is the case spread some rags around the brake-line that exits the M/C and crack it slowly.
Caution-1: When you do this; liberally spread rags around the area, cuz brake-fluid attacks paint. It's usually not instant so don't panic. If you get some on the paint; wipe it off ASAP, followed by a soap and water wash.
So lets say there is already pressure there, see note-4, but if not, see Note-5

Note-4
This can only happen if the fluid is not able to return into the M/C. Tighten the line, then check for pedal travel. If you now have travel, then the problem is inside the M/C. The first check is to make sure the Ports are open, down in the bottom of the reservoir. If they are, then see note 6

Note-5
To see if the M/C is pushing fluid; After having tested for trapped pressure and finding none;
have a helper gently apply pedal, then crack the line again.
Caution-2: when you crack that line, and things are operating correctly, the pedal will fall, so your helper has to maintain an easy pedal. If the helper lifts, then air will get sucked in. If the helper pushes, you will have a gusher.
If fluid comes out in this circumstance then the M/C is fine. But if the pedal freezes or remains frozen, or no fluid comes out, then the M/C is no longer working; either rebuild it, or replace it.

Note-6
Testing for a too-long pushrod
Loosen all the bolts that attach the M/C to the booster, about a quarter of an inch, pull the M/C off the booster and wedge it there. It doesn't have to be pretty. This will allow the pushrod plenty of room to park. Then go pump the pedal. If it now operates normally, then your pushrod is just too long or it is not seated in the backend of the Power-piston. Find out which it is and fix it; then retest.

Ok then, I think I covered it all......... at least the hydraulic side of the system.