There has been a lot of discussion on hunts, and there is an obvious consensus that they are not functioning as intended and need to be fixed. There is less understanding of the root problem, however, and a lot of nice ideas that don't quite hit the mark, and other ideas that will actually make the things worse (e.g. "Make monster invincible for x minutes so everyone on the server can gather to kill it").
The problem is that there is way more participation than the system was designed to handle.
The root cause of this problem is simple:
The rewards for killing Elite Marks without a bill are way too high.
Right now, hunts are basically Walmart Black Friday sales for gear. For any Japanese people who will (hopefully) read this and don't know what I mean, on Black Friday, the biggest shopping day in the US, Walmart will offer a limited amount of highly desired items (like HD TVs) at a very low price. This creates huge mobs of frenzied customers who compete for the items. Almost every year, the mobs will literally trample someone to death.
Most players will grind the content that gives then the greatest reward for the least time and effort. Game balance issues aside, as long as hunts are the most profitable and easiest activity possible, participation will remain very high.
Rewards need to be restructured such that they are more in line with other content. Some players will whine, but it is more simple and feasible to reduce hunt rewards than to increase everything else. Also, the sooner this is done, the better. Given the nature of elite marks and how they are designed for small groups and limited in availability, the reward structure should discourage over participation by the same people so that more players can experience the content. Hopefully, doing so can reduce demand to the point where people can experience hunts the way they were designed to be experienced.
One other change that should be made: Elite Mark Bills should be randomized or handled differently. Assigning every player in the game to one monster guarantees that they will be mobbed like they are now, and that the weekly big bad monster will never be more than a big piñata.
I think the hunt system itself is fine and could be very enjoyable as Yoshi envisioned it. The problem is hunts are like beaches: they are ruined by overcrowding. Fixing the overcrowding removes the need to make most other changes people have suggested.
The other way to fix hunts would be to fundamentally restructure them in such a way that they do not break with over participation, such as instancing. Since that is a significant departure from the original vision for hunts and will take much longer to implement, adjusting rewards is probably a better route.
EDIT:
Based on some good discussion on this thread, I believe that other content should have rewards raised and limitations eased simultaneously with reduction in hunt rewards to prevent stratification of the player base into two gear levels based on who exploited this and who didn't.