The problem with the suggested approach is that the bus and psv keys are not synonyms, nor are the values yes and designated. PSVs include vehicles other than buses and there needs to be some evidence that the access restriction or exemption *really* applies to all PSVs and not just buses.

If busmiles doesn't work with the bus tag alone, that is a problem for their developers rather than a reason to break OSM. You are not the first busmiles user to add psv tags or to misunderstand what motor_vehicle=designated means.

People use OSM data for real-world routing applications, so misapplying access tags can cause more significant inconvenience than failing to snap a bus route to a map.

Bus routes can be accurately represented in OSM using route relations. It would be a lot more helpful if busmiles used these rather than interfering with access tags which people have taken the effort to map and survey accurately. e.g. Bus 1C: Newbury Wharf => Thatcham => Newbury Wharf
https://www.openstreetmap.org/relation/17716750