I bought one of these fine old Browning designs in .35 Remington, only to find the primers untouched when I tried to shoot the thing. Was pin breakage an issue with these guns?

Not uncommon for the firing pins on Model 8 Remingtons to break. Not uncommon to find home made replacements inside either.

The bolt comes out fairly easily to get at the firing pin.

Make sure the rifle is cocked.

TD the rifle

With the rear half in hand,,pull the bolt rearward about 1/2" and hold it there with your other thumb.
(Do not pull the bolt all the way to the rear and have the bolt catch lock it into place. If you do that, you'll have to re-place the bbl assembly into the receiver to unlock it)
Pull out the small knob plunger in the center of the operating handle and slide the op handle forward and clear of the bolt.

Now the bolt will be free to come out of the front of the receiver. It's under some spring pressure (as your thumb knows), so don't let it take a ride. Protect the end and catch the bolt as it slides out the front.

At the rear bottom right edge of the bolt (looking at it from the back) there is a pin.
That is the firing pin retainer (pin).
Push that out from that end. It has a flat ground on it so it only comes out and back in place one way to still allow the FP to work freely.

The fireing pin will come out the back of the bolt.

On the earliest Mod 8's, there were no firing pin coil springs around the pin to retract it. Later 2 springs were added, an IIRC not at the same time.
A short stiff one at the rear called a F/P Buffer spring. The longer smaller in diameter one that surrounds the long thin nose of the fireing pin was simply called 'firing pin spring'.

The shorter 'buffer spring' will come out with the firing pin. The longer 'firing pin spring' in the forward section of the bolt assembly (the bolt head actually) will stay put IIRC. You have to take the bolt head apart from the rear half by removing the cam pins. It's self explanitory once you have it in hand. They are pretty simple actually.

Early style bolts have no room inside for the later installed springs. I imagine they could be modified and Rem probably did. But it's not something the average gunowner would get in to.

A broken pin may leave a piece or pieces inside the bolt to be fished out. Home made pins sometimes bend and jam sideways allowing the misfires.
The worst of the firing pin faults is when a piece of a broken pin or a bent pin jams forward and causes the rifle to cycle full auto as the bolt is closed on the first round.

I have a few firing pins for the Model 8 & 81
I wil be glad to give you one.