... without requiring a recreation of an improbable event in all of them

Just to add to your post: the probability of such a spontaneous event (life emerging from a cocktail) increases as you're no longer looking at a restricted area. If panspermia is the mechanism then you don't require life to originate on a suitable planet in a Goldilocks zone. Every square mm of (often porous) rock flying around is a potential point of origin, carrier or seeding station.