that's actually not true. At various points, there have been both male and female dominated societies. I am too lazy to go and write a whole 'nother essay and dig up references, but you can go and look it up if you want to.
but to specifically answer your criticisms about the state of western liberal society and it's satellites such as Singapore, what you describe I would term as societal backsliding. It is not a problem of women dominating. It is about losers dominating. You can end up in nonsense very quickly as you point out. A convenient political description is too far to the left. There you only have to shut your eyes and your wallets to all that nonsense. Life still goes on.
But to go to the other end with only a single power dominating is usually just as bad or perhaps worse. The Eastern European societies you describe as role models are no different in state control of the individual with present-day Muslim dominated theocracies. The only difference is that instead of following a religion purporting to represent God, they follow a religion purporting to represent the people. Either way the answer is the same, a few people control and lord over the lives of the many. I would not like to live under this.
To criticize Islam while supporting the Eastern European control model is frankly nonsensical. They are functionally identical. Just replace colours, idols, flags, etc. and you have the same product. Muslim countries will happily point out that they have their "morals intact".