What you are experience is not uncommon at all. Many, many people have had these exact symptoms.
Pilot screws all way in and still being able to start is sure sign there's internal issues with carbs. Possible jet-drilling by previous owners. And/or corroded/crumbling pilot O-rings. Did you replace those?
Also leaking fuel-rail O-rings will suck in air and cause random variations in AFR. On my race bike, this caused AFR to be lean on one cylinder on left turns and lean on other cylinder on right turns.
Ignition issues do not vary. If you have weak spark which causes sputtering and poor running, it will exhibit these symptoms full-time, from first start onwards. If too weak, it won't start at all. But will not cause running for a while and stopping.
Your issues with dying out sounds like fuel-flow issues. Test petcock-flow rate and actual vacuum in hose going to petcock. It's not uncommon for diaphragm in petcock to harden, crack and leak over time. Causing insufficient activation and low flow-rates. While bike sits, it will actually leak fuel slowly into carbs and fill float-bowl. However, when bike starts, fuel-consumption is higher than leaky petcock is able to deliver and float-bowls run dry and bike stops after about a minute (exactly 1/2 lap at Thunderhill, wonder how I know?
). Leaky vacuum hose between carb and petcock will also cause similar symptom. That's why you'll want to measure actual flow-rate from petcock to rule it out as contributor to issues.