I don't see much difference between a stairlift fitted in an auditorium or a stairlift fitted in someones home. Indeed, as they are usually fitted in a standard stairwell, they pose an increased level of risk as they reduce the overall space that probably wasn't designed from scratch to take such a device.
Indeed, the auditorium would still provide the same access or egress as that provided previously - depending on where the lift is sited, of course. Naturally, they shouldn't be placed in the main centre aisle - but along one of the walls.
Climbing is much harder for many mobility challenged people. At my local Vue cinema, only one of the auditoriums has mid-level access - all the others have bottom level only, and all fail to provide even basic hand rails.
Plymouth's Vue cinema was refitted not so long ago, probably within the last 7 years or so, so I find it difficult to align your comments with my experiences.