Replacing a tub usually involves more demolition than just cutting out the old one. Often at a minimum, the lav/vanity and/or toilet will have to be temporarily removed. You are correct that you cannot stand the tub up vertically and then tilt it down. You must bring it into the room in a horizontal mode, although it can be turned so the front apron is up. To have clearance to move it into position, some drywall, door jam mouding, etc. may have to be removed.