Rebuild or replace our chimney?

Is it better to rebuild a rotted chimney frame for our current gas log fireplace or replace it with a dog house chase and direct vent fireplace?

It would certainly cost more to replace a rotted chimney frame. However, keep in mind that even if you use a direct vent, you still need to repair the space where the chimney is now. Even if you take it down you will need to fill in the space with whatever material is on the house now. So while a direct vent is cheaper, it sounds like you will be spending money elsewhere.