I agree that many users would not know how to set up their own server, but would want the functionality it provides. I think ultimately there would need to be a "server" installation of MMEX. That should be fairly simple, an installed for a SQL DB of choice and the sync component. For the initial phase though I would say implement the sync and let people with can set up their own server try it out.
In terms of support I think you would find this requires less support than the current system, not more. The issues you're having with dropbox sync is that dropbox was never intended to sync database files in this way, trying to use something for a purpose it was not intended for is always going to cause issues. By implementing something that is designed for this purpose, once the bugs are sorted it, it should work fairly smoothly.
I appreciate this is potentially a fair bit of work, but I think it's worth it. I like MMEX but there are numerous other free open source solution out there that do a similar job. The thing that set MMEX apart for me (and why I'm here) is that I need the ability to enter transactions when I'm out of the house. That's when I make most of my smaller transactions, and currently regularly forget to record them when I get home. The android integration you currently have sort of resolves that issue, but there are too many limitations. Not least the fact that if I forget to close the desktop version and enter a transaction on android I'm going to end up with 2 versions of the database that are difficult to reconcile.
Take a look at the SymmetricDS site and see if you're happy with it and what it can do (there are other similar solutions out there but I think this is the best fit for this project). I believe if you get this right it will really set MMEX apart from the crowd.