I actually think they do play, but anonymously. @ZOS_GinaBruno talked about a character in a blog or something, but said she would not reveal the name. However, I've played since beta and have never once encountered a game master in game.
Lag is ruinin' my 'mershun! A society grows great when old men plant trees whose shade they know they shall never sit in. There is only one good, knowledge, and one evil, ignorance - Socrates Member of the Old Guard, keepers of the game's history
I think staffmembers can have three reasons to play.
1: to take official actions against other players.
2: to test things out like potential bugs.
3: just cause they like to play.
For the first they will use the tag. But for the others, it is actually better to turn it off (yes they can turn it off). When testing things they don't want put attention to them cause that can ruin what they are testing. when they are just there cause they like to play, they are not working and so don't want to be talking with players bout the game in an official capacity.
So it is very unlikely you will ever see a staffmember ingame. at all, and if you do, you likely have been naughty
Through me you pass into the city of woe:
Through me you pass into eternal pain:
Through me among the people lost for aye.
Staff members would never really give out their account info or names because they'd be hounded non-stop by players or reports or all sorts of general nonsense. I work for a social MMO and used to be Counselor for UO, and I tend to get tons of people messaging me asking for random things (money, bans, etc). It's fun to socialize, but the moment people find out you're staff in the game they treat you differently.