You’re going to struggle for traffic free, the only genuinely traffic free routes I can think of is along the don valley (5 weirs route) or there’s a path along the side of the parkway (road) into town. However these are sides of town that I personally wouldn’t want to live.
Scandalous has suggested the Hope Valley, which is nice but the commute could end up as a grind every day both ways as it’s pretty hilly (unsurprisingly), but you could use the train (Hope, Bamford, Hathersage, Grindleford) if you didn’t want to cycle each day.
I’d suggest a suburb rather than a Peak District village, best of both worlds IMO. Lodge moor, crosspool, ranmoor, fulwood, whirlow, etc on the west of the city has easy access into town and MTB on the doorstep into the peak. However the commute is not traffic free, the only part that is vaguely traffic free would be down through the parks Endcliffe then onto ecclesall road.