I dont think showing wartag its a good idea, maybe adding the choice to show it or not its better. If im with an alter i might wanna feel like playing with my new character, not feeling that im with my mains alter. Same way i remember at the beginning of wow classic a guy in my server who did a silly coment in the world chat, became famous in the server and everyone made fun of him. He had to delete and restart his character with another name, i guess that he wouldnt like the idea of being recognizable even then.
Un saludito a josenoseque que se llamaba el tío.
The idea of the WarTag is, when someone adds you in his friends list, to be able to see with which character you are currently online, and not to have to add all your characters in his friends list.
And if you need complete privacy (after all we all need it sometimes) and want to make an alt, you can always make a new account, the game is completely free :)
You will of course have to option to "appear offline".