When it dies and you hit the starter, do you also flip the choke lever to the half or full position?
I've tried it in about every position possible. Why do you ask?
I ask because I'm trying to understand exactly what you've done when this condition has occurred.
A local K75 presented similar symptoms. The bike would start and run well until sometime later in the day when it would shut down at idle. The operator would give it full "choke" and it would start. He'd shut off the choke and the bike would start and run well the rest of the day. This problem and its remedy occurred on a few consecutive days.
I removed the fuel cap assembly and fuel from the tank to look for debris. I found a relatively clean tank interior but the fuel pump seem mounted too low in its holder. Figuring that if the pump were somehow resting on the bottom of the tank, it might be causing an intermittent fuel restriction, I removed the whole pump assembly. I re-positioned the pump upwards in its holder so the vibration damper would grip it between the two beads on the pump body as it was designed to do. I re-attached the filter basket to the damper then remounted the whole assembly.
I've received no reports of the bike having stalled since that was done several months ago. I saw the bike flying down I-89 the other day, Burlington-bound.
Your coolant hose should not look like that. If the bike is approaching overheating, the fan should be running. The overheat lamp should be lit if the fan doesn't run and the bike is overheating. Eventually the cap should blow to release pressure if there is acute overheating and no relief. You should consider replacing the cap, replacing the hose, checking the thermostat, flushing the coolant system, replacing the coolant and checking the operation of both the fan and the overheat light if you haven't done so already.