Well, I don't know why a push button is needed but you need to check and see if you are getting 12VDC at the starter solenoid. If you are, throw the starter in the trash and buy this,
https://www.ebay.com/itm/PREMIUM-NEW...lXBAuq&vxp=mtr
If you are getting less than 12VDC you need to check your wiring for issues.
Do you know why the po installed a push button? If the switch isn't working properly get a replacement as here.
http://www.messicks.com/part/66706-5...gnition-switch