I know some games have a "gear score", which is basically the sum of the scores on your equipment (we don't have scores in this game but something equivalent could be found. For example: the level that the gear can be equipped at without reductions + the sum of the upgrades on gear = gearscore. Probably need a more complex formula than this, but I suck at math.
The qualification for entering an dungeon on a certain difficulty, can be a characters gear score. Also perhaps have a Party gearscore, which would be the sum of all gear scores in the dungeon party, this would not be used for entry, but some people like statistics. I guess there could be a score at the end of the dungeon, based on how fast the dungeon was cleared, compared to others of similar party gearscore, and extra cursed fragments awarded based on this.
Another UI modification, needed so people know their gearscore, would be to have the gearscore value, simply printed at the top of the character statistics under H, that would take minimal coding.
I think this suggestion, is a nice middle ground, which is fair to people wanting to run dungeons to get better gear, and also fair to people who want to run dungeons as fast as humanly possible. Mid to low gearscored people will run with people with similar scores, and the higher with people of similar score also.