Not a terrible suggestion, but I want to point out that HH currently has the benefit of bringing actively-online players together at the same time in order to run dungeons (HH requires a team of 3+ for the bonuses)
If Lava changes this to be "Your first x runs per day", then you start to encounter the issue of not having people to run with - standing around the dungeon area, waiting for people to join in. This could easily take hours if you're not spamming the "find group" button to teleport you to other servers where people are gathered.