... developing it's own API for Windows-based games that could be superior to WINE...
This is an extremely bad idea. WINE has been around for a long time, the developers behind it are already busy at work, the project is not going to be stopping soon, and it is already in high use by the community. Opening an entirely new project to do the same thing goes completely against the ideology of open source and the idea behind the GPL. You might like to take a look the Wine Doors project (http://www.wine-doors.org
), which means to make using WINE and installing certain applications supported by WINE easier.
So, taking that into consideration, one thing that Linux Mint could do is give more support for WINE in the operating system.