I think by "kept in check" they mean good separation of church and state. Some islamic countries don't have good separation of church and state.
Some (most?) forms of Islam are just not compatible with western style democracy and secularism. It’s unfortunate but I don’t really see any other realistic options of solving this besides “oppression”.
Some countries like Jordan kind of partially pulled it off but they had very specific conditions which can’t really be replicated elsewhere.