Just an idea. I don't know what JH coaches do (I think Regent Centre & Four Lane Ends) on Plate Day but if we're likely to get "normality", would a shuttle from Gosforth Park to Newcastle after racing not be an idea? Obviously pricing would need to be competitive against a taxi but would save a walk to Great North Rd for the 43/44/45 as well as offering better links.
JH did do a shuttle on race days previously. Stagecoach and Classic have done it in the past too. It's up to the racecourse to see if the demand is there for a bus shuttle.
Depends if what JH do is free or not. If you've got a free or very cheap shuttle into Metro stations then it's not really worth trying to compete with it.
If people want to pay for taxis then they'll get a taxi, especially if it works out cheaper in a group.