I was at one time a certified structural steel and pipe welder. My welds would at times be x-rayed in order to pass inspection...not trying to impress you, just letting you know I'm not pulling this stuff out of thin air.

When dealing with joints that require structural integrity, Tig welds are not usually ground down for cosmetic purposes. All hand welders try to achieve the "off-set dime stack" look that you're talking about. If my Tig welds did not come out clean, I'd either toss the metal or break the weld, grind ALL the weld off and start over again. Needless to say, this was a rare occurance after a while. On the other hand, it is not uncommon to see stick welds (arc) grounded down, but there is a reason for that. Stick welds are more prone to leave voids.

You can grind the weld down, but use caution that you don't compromise the structural integrity of the weld. If you do grind any significant amount away, you may lay a new bead over the old weld, again that's not the norm with Tig.