Any Bank that wants to add Bitcoin their asset, shouldn't think of stabilize the market price because they don't have such powers, Banks especially the Central Banks can do well to controlling the activities and inflation of Fiat as that is within their purview, but they cannot extend that to Bitcoin. However, the reason why banks are currently investing and holding Bitcoin is because they've seen how potential Bitcoin can be, initially a lot of them were skeptical in adopting Bitcoin, else Banks would have been the highest adoptors of Bitcoin if they had the foresight.
Banks are really the ones that will enjoy investing in Bitcoin more because of trout long term foresight they always have when it comes to investing. But can they extend that to Bitcoin which pose a rivalry to them? Actually Bitcoin was only made to be an alternative to them but not to replace them but due to wrong narratives, they’ve also missed out to have invested earlier in this assets. After carefully studying what Bitcoin encompasses, I don’t think banks will be having much issues with adoption Bitcoin and allowing not to be traded through their services soon.