Yes, Pokémon Transporter and Bank still work.
In order to transfer Pokémon from Transporter, it first needs to be deposited in Bank, then put into your desired game from Bank.
(Crystal -> Transporter -> Bank -> Omega Ruby)
Don’t worry about the “free trial” or question marks: these are simply rendering issues from Nintendo making the application accessible in a way it wasn’t exactly expected to be (which they can’t fix if they wanted to because, no eshop) You have infinite, unlimited access to Bank and Transporter, until Nintendo decides to shut down the servers.