A whole lot of the usual worry about a straightforward Problem.
Of course asbestos anywhere needs to be treated seriously.
Without detailed knowledge however it seems reasonable to consider
the asbestos brake shoes for removal.
However understand that unlike car brake shoes their purpose is not to
bring the lift to a halt but hold in position when the brakes are applied.
So a lot less asbestos dust will be released.
The shoes will be either on a hauling drum at the top of the shaft or mounted underneath the car itself.
Either way it will probably take longer to remove any dust than to
change the brake shoes.
So no need to consider it a major problem, even less of a problem
than changing your cars brake shoes.