Actually it works now - just checked. The logic is as follows:
1. You need to have at least 5 wins before you retire and you will have BJJ and Wrestling unlocked;
2. You need to make another 5 wins in your next run to unlock the MMA as a style pick for your next run;
3. In MMA you need to select which styles should be allowed in your draft - by default you have all styles allowed and you need to tap those you don't want to be included. Next it will allow you to build your Deck by picking between two random cards for each slot (4 slots in total).