No-one says that the public transport at GP is "the bee's knees".... however, it did have one of the better dispersal rates in the country a few years ago according to the football research unit. It is served by far more buses than BMD, with many more access routes and available traffic lanes. Kirkdale is just as close to GP as Sandhills is to BMD and sees 2/3rds of the trains that serve Sandhills. Bankhall the rest. Unfortunately the northern line serves less than 25% of the city's population, which might explain why only a few percent of our home support currently use it on matchdays.
The key thing for BMD is its proximity to the city centre. At over a mile to all city centre bus terminals, it is probably at the edge of max desirable walking distance. If the few access roads are blocked for everything except shuttle buses then that could improve accessibility via public transport..... because as you have seen, there are not that many spare buses in the system for matchday soccerbuses, so the few that there are need fast turnarounds. The adjacent developments at Liverpool waters/Ten Streets will hopefully prompt the Vauxhall station to at least increase platform space and improve marshalling etc. A dedicated people-mover akin to the proposed Lime-line trackless tram could make a massive difference too.