My key fob is not starting the car unless it touches the push button. I’ve changed the battery and it still does this. The remote will unlock and lock the doors but it will not let me unlock the door from the door handle. Any advice is appreciated.

 

From your description your FOB key is in power save mode, one clue: FOB LED would not light, search in Ytube "Mazda key power saving mode".

 

Double check the battery voltage. Need to be over 3v, not 2.99 or 2.98.

Also there is a service bulletin regarding the modern remotes being faulty.