The FCC doesn't have jurisdiction over studios or their contracts with the networks, so they can't force studios to grant stations streaming rights to their content. If the FCC still somehow passed a rule requiring the streams to be a 100% simulcast of their on air signal or face a fine, the stations just wouldn't air that programming at all anymore and the telecasts would only be available on cable channels as they are not regulated by the FCC.
The Frosty/Rudolph streaming rights issue isn't a US only thing either. This past Sunday CBC had to blackout their airings of Frosty and Rudolph on CBC Gem's live streams and replaced them with The Great British Baking Show's Christmas specials as filler.
Also that's not what must carry status means. It's a provision that lets stations opt for free mandatory carriage, which is how local stations that show nothing but religious, shopping or brokered ethnic programming get carried, along with truely independent broadcasters that don't have a Big 4 affiliated sister station. Right now that provision only applies to cable, satellite and telco/fiber providers.