You still need to use an account that has purchased Arma 3 in order to download/update the server files. This makes it impossible to set up eg TCAdmin to be able to update the files without logging off the owner of the game from Steam in order to log Steamcmd in.
There needs to be a way that Steamcmd can download only the files needed for the server using an account that has not purchased the game. I am sure that this is possible, because that is how RO2 for one works.
Try updating the game with an Steamcmd (when there is an update available!) using an account that hasn't purchased Arma3 - It won't download anything and will tell you that the game is up-to-date.
Immediately afterwards try with an account that has purchased the game and it will update.
I have only tried this today with the Dev version since there isn't an update for the main branch at the moment, but it didn't worked in the same way as this when there last was an update on the main branch.