I had never played a character that did not go to Riverwood first so I assumed only Alvor taught it. My last character started out at another location due to Life another Life, Alternate Start mod so I went to Whiterun first. And I don't mean smithing trainer, I mean the smith giving the tutorial on smithing. Was very refreshing to get taught by someone else without a woman walking by saying "Your pretty, I'll give you that. Just stay away from my husband Alvor." after my character spent the day sweating from the heat of his forge fire and covered in coal dust.
And I always smith. Its fun and I like the sense of accomplishment from making my own armor from the stuff I find instead of buying it. So I always do the smithing tutorian and sell back to them what I made to pass the quest.