I've gone on the treadmill w/ the guitar a couple/few times. I had to walk slower than I wanted to. If at good walking clip, no accuracy and bad posture; i.e. back pain. Negatives outweigh the positives for me. Your results may vary.
Strolling minstrel around the neighborhood sounds better cause you will naturally walk at a pace you can play at rather than be forced a pace by the treadmill.