OK. So there should be NO fuel flow from gas tank no matter which of the three positions (on/res/off) the petcock is set on, UNLESS engine is spinning (running or being cranked).

Since I DO have flow, the petcock needs a rebuild. And I checked all three links you provided. What I noticed was that all three indicate incompatible with my 96 F3:
dennis kirk link shows compatible with f2. If I load 96 f3 into garage then search for petcock rebuild kit, there are zero results.
parts giant link same as above
amazon all ***** has notification "! This does not fit 1996 Honda CBR600 F3" (I was logged into Amazon and had previously loaded the bike into my vehicle garage).

After viewing the above kits I visually examine the petcock on my bike and notice there are no screws holding it together. It does not appear to be a serviceable part at all. And although I am not the original owner, I do know the complete history of the bike and can verify that petcock is original equipment.
What the heck?

 

So sorry, got your F3 mixed up with F2. It doesn't have vacuum petcock, so yes it's flowing properly when set to ON.

Fuel pump is common issue on F3 and many people have removed pump and used gravity feed direct to carbs. Give that a try.

 

Trying the gravity feed and so far no luck starting still. I am wondering if I may not have enough fuel in the tank? I was going to spray a bit of starting fluid, but could not see a direct path to the airbox or carb inputs to make this feasible without removing the tank.

 

Might try running 1/2" hose down intake tube to airbox. Then pour/spray 2-3cc petrol down hose. Wait 15s, then start bike.

Most likely carbs completely clogged and gummed up since we know petrol flows from petcock easily. Really difficult to restore carbs back to factory fresh clean. Needs completely disassembly to every last nut, bolt & individual component. Every secret hidden passage in carb-body scrubbed out with brushes and PEA-based fuel-system cleaner. Everything ultrasonic soaked and micro sofa-blasted. Replace all rubbers: float valves, pilot-screw O-rings, fuel-rail O-rings, float-bowl seals. Lots of work and needs specialised equipment.