(two, actually, for redundancy) And 99% of the time, they're both used at the same time. Especially when cold. So you get one start attempt.

hooked up to the N2 (high-pressure) turbine) Nope. The JFS is mounted to the ADG (Accesory Drive Gearbox), where all of the other fun stuff (hydro pumps, generators, etc) mount. Power transfers from the JFS, using a gearing system that changes the JFS's high-speed/low-torque to low-speed/high-torque, which then passes through a PTO shaft, which is connected by a gearbox mounted on the engine.

it's also directly connected to the engine See above.

double-duty as the brake system accumulators EMERGENCY brake accumulators. Brakes are normally fed by the dual hydraulic systems, which each have their own accumulators.

hydraulic accumulator initiated starters are very much something for the military - they're quite dangerous to handle It's a metal tube with a piston in the middle of it. Drain hydro pressue, deplete the nitrogen charge and at that point it's a hunk of metal.

only having a couple of cold start attempts You get one. And then....

repressurisation from an external hydraulic supply You see maintainers lining up in front of the left wheel to pump that fucker back up by hand. No starts are a "everyone gets a turn on the pump handle" event. Unless you're new or pissed someone off. In which case you do it by yourself. Why do it by hand you ask? Becasue the external hydro supply (called mules) take 15+ minutes to get when requested and don't always work when they show up. The pilot is waiting the cockpit, and when jets take off late, someone gets the red mark, and no one likes that. Manual labor always works.