They don't own their own rights. Would anyone be surprised if the Os try to claim the Nats share of an MLB package and then pay them for it via the masn agreement?
MLB streaming rights are hard to understand and I've never read anything about who owns the Nats and Os. MLB has made a point of keeping them separate from TV rights and weren't allowing any TV rights deals to include streaming rights except by waiver for tv everywhere type apps. Then at one point, MLB gave them to the team, but was annoyed when some of the teams sold their rights to Bally's, and put the brakes on anyone else selling them to them.
It seems pretty unclear whether MASN owns the rights other than by waiver for their tv everywhere app that requires a cable or equivalent bundle. If they did, they'd likely offer a standalone $20-30/month thing.
MLB has said they want to package as many of the local streaming rights together as they can, so wouldn't be surprised if the Os sale included language that the Nats and Os will be part of that, so it doesn't end up in a big legal battle down the road.