In Patch 7.0 Blizzard introduced stat templates for PvP. In B\battlegrounds now, you don't get stats from your gear. Instead your gear stats are overridden by a stat template. All characters of a given specialization share the same template. Your gear level does increase the amount of stats given by the template, but by a much lower amount than in PvE.
These stat templates apply to low-level PvP. I've been leveling a rogue without heirlooms and dabbling in some PvP. In my mind, these stat templates are amazing. Low level PvP is actually fun again. You don't get one-shot by characters decked out in heirloom gear. In my mind, whatever the impact at max level, the stat templates have rejuvenated low level PvP.
Then I did a low level dungeon. It was pretty terrible. The other characters were decked in heirlooms, so they just zerged the entire thing. Bosses died in less than 30 seconds. There was no skill or strategy, or any sense of group play.
The problem is that leveling dungeons need to be balanced such that a group of new players in quest gear can complete them. But if one or more heirloom characters are present, that balance goes out the window.
I think stat templates for leveling dungeons would be a great idea. Everyone would be reduced down to an even playing field. Dungeons would be a proper group experience once more. I rather doubt anyone will sheep anything, but maybe it could happen.
Heirlooms are fine for solo-play. They can be overpowered in the world. But when playing with others, I think it's more important to provide a fun, balanced, and reasonably challenging experience. Heirlooms will still give the character more XP and a high ilevel, but at least with stat templates the disparity and zerg would be greatly diminished.