Larger surface area surely increases the risk as there is more area that could run over glass shards.

I have 700x35, or 38, can't remember. Had 1 flat on the front wheel in the entire lifespan of the outer tread. Second outer tyre now and nothing so far.

That mKes it more likely to get flats though.

Meaning more surface area to hit something sharp. Yeah the larger area means lower pressure on the contact surface but I don't think that's the deciding factor in whether something punctures your tire. Best way to prevent punctures is tougher tires rather than larger ones. The lower pressure just helps with stuff like loose gravel or sand.