The per-line timestamps in plain-text logs are turned on by a hidden setting. To disable them:
1. Go to Settings > Advanced
2. Search for timestamps (or the exact phrase "When logging plain text, include timestamps for each line?").
3. Set that setting to No.
4. Restart logging (stop and start it, or start a new logged session) so the change takes effect.
The "Timestamps" checkbox on the profile's Session pane is unrelated. It shows timestamps in the terminal window, not in the log file, so leaving it unchecked won't remove the timestamps from your log. The Advanced setting above is the one you want.