Importing subtitles from MPG files and burning them into an exported new file

Yes. That would be a feature request. Shotcut can do it, but not automatically.

The steps to do it in Shotcut would be:

  1. Open the file in Shotcut
  2. Extract the subtitles
  3. Add the clip to the timeline
  4. Import the subtitles in the Subtitles Panel
  5. Add the Subtitle Burn-In Video Filter to the output
  6. Export

However, in your specific case, you can not do step #2 because your file has captions, not subtitles. So you would need to extra steps to convert the captions to subtitles in an external application.