I certainly endorse the recomendation that you instal a separate switch to control the operation of the starting carb but would add the following:
Another component that can adversely affect the operation of the starting carb is the anti-blowback valve, located at the base of the intake manifold where the starting carb mixture-pipe enters. (Removal of the large banjo-bolt gives access to the spring-loaded valve.)
That valve can become gummed-up from normal gasoline residue but the situation can also be aggravated if the inside of your gas tank has been coated with a rust preventer. Some gasoline additives will gradually dissolve that coating and serve to cause the float-bowl valves, the starting carb solenoid-operated valve and the anti-blowback valve to regularly stick closed. Just sitting overnight may be long enough to cause the problem or it may take a few days after your last outing.
You can give the float-bowls a friendly tap to free their valves but the solenoid-operated valve and the anti-blowback valve will require some minor disassembly to clean them.
If you suspect that a gas tank coating may be the cause, the only solution will be to remove the tank and have it cleaned.
Good luck,
Dick