Assuming that abiogenesis wouldn't happen very frequently (given the right set of conditions, a mean time to happen on the order of thousands to billions of yea

Assuming that abiogenesis wouldn't happen very frequently (given the right set of conditions, a mean time to happen on the order of thousands to billions of years), a single life-forming event would have the same effect, no? After a foundation is set, the self-replicator starts exploiting existing energy gradients, making any other abiogenesis-event much less likely.