Your description of the starting sequence seems off. Usually its induction fan, hot surface igniter, gas valve. if the flame is not established in a few seconds, the gas valve closes & 15 or so seconds later the induction stops. If the T-stat is still calling it'll try 3 times total before it locks out.
If there is no crack in the igniter, try jumping terminals on the pressure switch. It has a rubber hose going from the induction blower housing to a diaphragm looking canister with two wire connected. While you're at it you may want to look at that hose before going to the trouble of pulling the igniter.
The pressure switch establishes draft before the gas valve opens.