The "pull-in" winding on the solenoid draws 50+ amps for a fraction of a second before the starter motor starts turning. Most people don't use a push button with enough amperage capacity to handle the surge current. Try a marine-rated starter button with at least a 50 amp momentary surge capacity. Another method is to use a Ford-style magnetic solenoid to power the "S" terminal of the solenoid. That takes the load off a light-duty starter button.
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!