If I'm not mistaken, modern sets go thru the menus to get to convergence. Seems that big screen sets have a menu that the user can get to and perform convergence, and from what I've seen, it's quite easy and just about anyone can do it as long as they have decent vision. I made some convergence adjustments on my neighbor's big screen a few years ago... it was rather easy. Certainly came out better than a roundie!
I believe smaller sets have a secret menu that is not available to the set's owner. If you bring the set in for repair, the repair man knows a certain sequence of buttons to push to bring up the service menu... and I would think convergence would be in there.
We've had discussions here before about the secret service menu... they were pretty interesting.