Negation of Advantage
At our table we use the Variant Flanking rules, however we noticed the same problem as yourself, and added the corollary that you cannot gain the advantage of a flank if you yourself are being flanked. The thematic justification was that you are busy trying to cover your own back and can't put all of your attention onto exploiting the enemy's defensive gaps.
This is justified mechanically under the rules for gaining advantage and disadvantage:
The GM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.
It led to players holding formations, covering each others back and using the terrain more to their advantage, using low walls and pillars to block enemy movement into the now more limited flanking positions.