Pressing the lights is only a self test of the bulb. It does not "reset" anything.
Normally you should only have to toggle the interrupter switch to the CLOSE position for a moment for the contactor to latch closed.
Could be an intermittent connection somewhere, a flakey contact in a number of relays, a loose relay, an intermittent connection inside S7 (the contactor control switch), corrosion in the K1 contactor, or possible other issues.
Glad it's outputting power for you!
There are ways to "hotwire" it but due to potential liability issues I would not want to publish the methods as it would defeat many of the safety systems in the generator which could put you and your loved ones at risk.