Originally, I recall that part of the impetus for the jam was to try and drive participation in the actual Q&A portion of the site (i.e., the main focus).
While this was a laudable goal, I don't think it actually ever happened. I can't find evidence of increased Q&A activity or quality during the period where the jams run. I'm guessing that the short time period of the jams generally tends to make people stay heads-down on their projects, working around problems or sticking to things they know very well in order to get as much time making a good jam game as possible.
So I don't think there's a particularly compelling need to keep the jam hosted on the GDSE meta. I think it should be moved to an external site, because that would solve a lot of problems. The minimum reputation is an issue that would go away, as is the fact that meta is not really suited to the presentation we'd like for the jam and thus requires some ignoble hacks.
Of course hosting the site externally does present some challenges, but I don't think they are insurmountable or really, particularly different from the challenges we have now on meta.
Somebody would have to host and maintain it. But somebody (John, and previously Alex) is already doing so, and all the hacks we have to do to the jam posts require the intervention of diamond moderators. Hosting and domain registration would be the only real new costs/responsibilities of a single user if we moved the jam, and it's possible to get those cheaply enough that the site could be ad or donation supported. Alternatively, using the various jam-service-providers mentioned by others in this thread could alleviate that problem entirely.
The only major issue would be the software to run the jam itself. This could be built as an open-source project to distribute the ownership and thus avoid the problem of a single user building it and getting bored and abandoning it. But again, there is already software to do this so we don't need to build our own at all.
I think we should try to move the jam off-site. It would mean abandoning the StackExchange name and all association with the site (other than linking back here, of course), but I think it would be better for the jam itself and would allow it to grow much more than I think it's going to if we keep hosting it here.