I think people are busy reading into his tweets and anything else that means nothing at all when they should listen to his comments and pay attention to his pitching.
He's not a free agent for another three years. There is a lot of angst here over something that might not be an issue come July. Or 2022. Or 2023. I seem to recall a lot of angst about Michael Wacha, NLCS MVP, going to arbitration and what that meant for his relationship with the club, and then none when he left as a free agent. Things happen. No reason to read into something now that isn't there now and may not even be a consideration later.
Flaherty has never articulated a wish to "go to the west coast" any more than any other player has talked about playing near his home or going where the weather is warmer or spring training is closer and on and on and on. He lives there, so heading back to LA for the winter isn't a message. It's home. He has talked about the system, and how he would like a chance to make what he feels his performance is worth. Is that going to happen in St. Louis? Well, maybe not until he's a free agent, because again that's the system in the current CBA. And three years from if he signs a free-agent contract with a team out west is isn't some validation that he was angry with the Cardinals over this arbitration hearing. It was likely that the team out west won a bidding war the Cardinals didn't entered, and the geography was appealing, too.
Here's an idea: Watch how Flaherty is treated on social media. If fans want Flaherty to call St. Louis home, maybe take up the cause there...