Manchester to Leeds is much more likely. Same mileage, more potential for traffic.
More likely but as the crow flies the centres of Leeds and Manchester are over 35 miles apart. Manchester and Sheffield are 32. The crow would fly 29 miles direct from Leeds to Sheffield in this northern triangle.
The Leeds to Manchester route is the favourite because there are other built up areas it would also serve although that means more twists and turns to thread through and around old infrastructure and any stops will delay end to end journey times. A 30 minute aspiration will be hard to achieve.
Compare that with Sheffield-Manchester. No setlements near the straight line to be served. Direct, level route would be all underground avoiding, once built, environmental conflicts through the Peak District and in built up areas.
The reason current routes aren't used anywhere is due to levels of current services, and perceptions of distance. Make it 20 minutes from city centre to city centre and it's very different.
However I've no expectation that this idea will get taken forward. The cost would be enormous, although I'd contend that the long term benefits would be a rebalancing of the Northern economy. But we'll be arguing about the details of Leeds-Manchester electrification and services for at least another decade and the final result will remain a hybrid old /new mix of lines and infrastructure woven between intermediate built up areas.
However the North /South divide is nothing to the rivalries within the North! Rivalries that thrive on competing transport schemes, and have done for centuries.
Of course a city centre to city centre journey time of 20-30 minutes is little benefit to me when it might take that long to get to the city centre to start my journey to improve on the 50 minutes it takes me now.