I'm not sure you could use New Station Street as the construction area surrounds the exit road to City Square, so buses would have to somehow turn around and head back to Boar Lane. Plus the section closest to Boar Lane is being re-laid as a semi-pedestrian configuration, and pedestrians have to walk on part of the roadway during the works which would make it dangerous to run buses along.
The two stands at the City Square end of Boar Lane are really the closest to the station that don't involve trekking through the active station or all the way around the construction works to the Hilton. Failing that the stands along Wellington Street would be likely the next nearest.
I should have made clear that I meant after the work has finished. Apart from that, you're right - the pavement on the left as you go up New Station street is being widened. This seems a little odd, as the bars are on the right-hand side. Mine not to wonder why... The Boar Lane stops are too heavily used by regular services for rail replacement (as is the one on Neville Street now, tbf), putting them behind the new loading bay on Wellington Street makes more sense. Huddersfield has a similar RRB trek, for some reason they don't use the former FreeTownBus stop.
Lack of signage has reminded me that late last year while walking to one of the pubs under the arches with a colleague, we met someone who wanted to know how to get into the station. If people can't even see the illuminated Southern Entrance entrance, I take the point that they might never find the replacement bus stop (and we did check that they just wanted to catch a train, not visit Boots, Smiths or the ticket office).