If a person were capable of self-rule and participatory democracy in a free republic, they wouldn’t be a muslim.
Participatory democracy presupposes that different people can have different goals, and that they can negotiate with others to each achieve their goals. In Islam, under Sharia, everything is already laid out, and there is no discussion or negotiation. There is only the determination of what the Sharia Rule is.
Self-rule is contradictory with Islam. The word "Muslim" means "slave of Allah".