You're aware of chimney caps and bug/spark screens, correct? Gas logs that vent through the chimney have two pipes; one for air intake and the other to exhaust the gases. They run vertical, parallel to each other. The front of the fireplace is sealed so there's no downward draft into the room. They can have an ignitor instead of a pilot light.

Look around, I'm sure you can find a model that meets your needs. Start here --> Heatnglo

Thank you, that was very helpful. I've been to multiple stores that never explained the intake/output pipes to me. Unfortunately, no one seems to think the chimney cap will survive the coastal wind where we live, with the damper sealed open. I may continue to look around for another store. I've been very happy with my vented logs at our main house.

Vented or Ventless gas logs do not have vent pipes, they vent through the chimney itself. A direct vent gas insert would use two pipes as described by @jrb451. With that said, an insert is likely a better choice than a gas log to resolve your issue.

Post some pictures of the fireplace and I can show you the difference of how they look.