No, mosques are vessels in teaching said schools of thought, which is why you have mosques, or rather imams, who declare them selves as from a particular school of thought, for example a shafi imam, or a amedi imam, or the obvious one, a whahabbi imam.The schools of thought are made up of people who study the vast texts of Islam, not just the Quran, and the Hadith, then determine what applies to Muslims.
To your second point, maybe not in years past, before the turn of the century however the situation is becoming worse, so I personally would like to see a move in this investment of funding of positive, tolerant, progressive, workable, "liberal" views and ideologies of Islam.
As Islam grows more and more in the west, this would be a good long term investment, and a way to secure the European or national way of life.
I'll have to concede at this point, but Russia(!) does this with religion and it worked out well so far, from what I recently researched. Despite the enormous checyan and Islamic problems they had, eventually they invested in Islamic schools of thought, booted out the radical, dangerous, cancerous schools of thought. What came about was a mutual beneficial relationship between the two parties. I certainly won't defend thier actions and human rights records, yet Russia has had a major history with radical Islam. Rather then go full trump, and yes I am aware of the wars that happened there, they tried a different tack, eventually.
