blower motor only starts after I turn power to it off and back on
Often, when the solution is to reboot it... that's an electronic problem. Something like that could be anything from a safety (temperature?) switch to a microchip error on a cpu.

Since you have a brand new motor, the installer should cover the repair, or know more about it. Usually work is warrantied. 

Obviously, someone needs to trace the electrical system. If the installer won't do anything (for whatever reason), you still probably want to call an HVAC guy before calling an electrician though, as it may be a problem that they've seen before (especially with motors that have been replaced).