If we could elect Ministers directly and replace them at will, wouldn't it solve the problem of controversial Ministers' decisions provoking strikes? It can be applied to pretty much every country, especialy France.
Not necessarily, because a (big) minority could be disappointed by some minister's decision, although the majority still supports him.
In general, there is a tradeoff between how representative and responsive any political body (in that case, the government) is and how good it works. If any member of the government could be voted out of office at any time, how do you get the system running? Long-term planning, bargaining and concessions, especially in multi-party governments, would be made harder.
Anyways, in most political systems, ministers are member of parliaments and could be voted out of office that way. Also, ministers quite often resign when there is huge public criticism of their behavior.
Ministers and ministries are not independent mini-governments. They are appointed by the governing party or coalition to implement its programme (as made into law by the parliament) and can usually be forced to resign by either the parliament or governing coalition if they deviate too much.
Ministries are also interdependent parts that can't function own their own, or have no purpose on their own (Finance Ministries). You would need to give all of them the power to set budgets and raise taxes to make them independent.
If ministers were elected directly with their own mandates, the country would become ungovernable. A simple (and likely!) example would be a Health minister being elected on a promise of free health care for all alongside a Finance minister elected on a promise of eliminating taxes. Conflicting priorities would cause an unending gridlock where none of the ministers is willing to compromise on their promises.
As a note: I expect that the "controversial" decisions that cause strikes are actually made by the government as a whole, the ministers are just the public face on them.