Basically they need to do it like ASUS did with their ROG phones.I
I personally don’t think that Sony's service is the only reason...after all, it's Japan only and Japan has a tiny market for streaming services (over 70% is physical media sales). Honestly, though, Sony should really get all their teams on the same page. Imagine how great it could be if the Xperia division did a collab with other divisions- eg. Xperia Alpha with great camera quality, Xperia Bravia and Xperia Walkman for content consumption, Xperia PlayStation for gaming, etc. (Credit to Sony Reconsidered for the potential collab ideas)
IMO you're right, they can rely on their other branch's already worldwide famous name and used it to market the phones.
As long as they take the integration seriously, it'd attack the fans of the already existing niche product and put more eyes into it.