Removing bedrock will be much easier than removing those bridges to the south! Have a look at the levels - they are already on a slope, so I don’t see how you would make them higher.Capacity at Sheffield has been discussed here before, and any improvements to the north are going to be very costly. (likely in the 'never going to happen' category)
So a possible (simpler?) solution, London - Sheffield services and Nottingham - Liverpool services both enter and depart to/from the south. Could elevated bay platforms be constructed at walkway level at the London side. Its a falling gradient into the stations, so would actually be less gradient. There is the issue of a couple of overbridges close to the station throat, but if that could be addressed would it help? Advantage is you wouldn't be removing large amounts of bedrock.
This would free some platforms off, but I agree does nothing to increase capaity going north, but getting the two EMR routes separate from Northern, XC and maybe Hull Trains might help
I’m not buying that four tracking to Nunnery is prohibitively expensive. Several of the bridges arent needed any more and it’s a tatty area. Dig it out, four track it with a lovely faster more flexible station throat, then redevelop the area, possibly over the line - it’s a site near the city centre and station so tidied up it would be worth a few bob.
Just needs a case for the gains in capacity and co-operation with the council over a masterplan (got to have a master plan with pretty pictures and flouncy words these days)