And if you don't let them, then it's not exactly a democracy, is it? It's foreigners forcing their values on the country.
Screw democracy, western secularism is objectively superior
From what I've read about post-colonial or communist states from WWII forward, before a stable secular and democratic state can be made, it requires international/domestic security, a successful industrialized economy, above-average education, and political sophistication that satisfies the people's beliefs without appealing mostly to religion or ethnic conflict. If any of these aren't met by the first 20 years or so after independence, the state is usually doomed to flounder in cycles of dictatorship and corruption that is extremely difficult to recover from. Secularism isn't necessary to fulfill any of those requirements, nor is a functional democracy (in fact it's often been a hindrance), but if they're all met, suddenly ideas such as secularism and democracy have real natural appeal and material benefit, which is what's really needed for the people to want it enough to demand and hold onto it.
The difficulty is that these countries are dominated by autocrats who devote the vast majority of their energy to the first item in that list, and typically have no motivation to develop education and the economy when the state can fund itself on oil or mineral exports instead of a developed economy. The result is that a large disgruntled populace is left with 19th century political ideals of simple nationalism and religious morality, making democracy unworkable (even if the people in charge wanted it to work), let alone a modern secular democracy. I also wouldn't read too much into those figures about most muslims wanting Sharia law; muslims who live under the Islamist governments that seriously implement Sharia law instead of just paying lip service to it sing a very different tune. It's a bit like people voting for Brexit and then regretting it, only replace "voting for Brexit" with "getting murdered by the Taliban".