To make sure I understand what you have so far:
-5 electric baseboard heaters
-each with their own thermostat
-powered on the same circuit
-by a double pole 30 amp breaker
-baseboard heaters quit working from time to time
-breaker never visually shows as tripped
-when you toggle the breaker 'off' then 'on', the heaters work for a short spell
Do I have this right? If so, here's what I'm thinking:
The only thing all your heaters have in common is this breaker, and the cable coming from the breaker to the first thermostat in the chain. I'm suspecting the breaker at this point. More specifically, its connection onto the buss rail of the panel. This Murry style breaker you've pictured is somewhat famous for developing a bad connection to the buss rail. This will effectively cut off one leg of the power. My thinking is that turning the breaker off then on "rocks" the breaker just enough to re-establish the connection on the buss rail to let it work briefly. My next troubleshooting step would be to unclip the breaker from out of the panel and examine both the buss rail under the breaker and the mating portion on the breaker for burnt spots. If this is what you find, replace the breaker with a new one, abandon this spot in the panel, and clip the new breaker in a new spot.
"Nothing is foolproof to a sufficiently talented fool"