Have you checked that your computer is receiving MIDI information from the G70? Do you have a normal sequencer to check this basic function?
Does your MIDI interface (or are you using USB?) have lights that flash to indicate MIDI reception and transmission? Are they working?
You say it 'seems' the MIDI connection is good, but are you sure the G70 is sending any? Check the manual for the MIDI section (p.214). You may need to make sure the Keyboard Part you are using transmits.
If all else fails (which it shouldn't!) you could always record the piece to the 16 Track sequencer (or Arranger Recorder) and transfer the saved file by USB to the computer, then open in Finale. But you shouldn't have to go that far...
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An arranger is just a tool. What matters is what you build with it..!