Possible.
I'm no expert, but it seems much more likely than interstellar panspermia, especially since we are dealing with a known source of living material over a known period of time.
Very early forms of life were anaerobic, and maybe they were more resistant to radiation (no ozone layer) than later, more complex forms.
If we discover any organic life in our solar system, the similarities between it and our own would be the first thing scientists would look for, and unless you believe in "parallel evolution," similarities would tend to support the "Earth-sourced" theory.