The irony here is that even if we had an uncorrupt religion within the state, the religion would quickly become a "football" on the political stage even in a democracy.
My view is that, in fact, religions should actively avoid becoming involved in politics. This is to religion and politics mutual benefit as it simplifies both. Of course, there is an overlap but I believe that overlap should be kept at the individual level for the most part.
Religious organizations can take action/make statements with political consequences but they should not endorse candidates. They should not seek out political opportunities to influence...they should just state their case.
Politics is about the work of convincing others to take action for the sake or good of the governing body (the polis). The power play of politics should be about drawing people to your view and being open to being drawn. Making deals and compromising when appropriate. Representing the interests of the people rather than merely special interests.
When you represent a diverse culture of religion, you can have religion for yourself but as a politician you should not project your religion into a political agenda. This should be a disqualification for politicians in my opinion.
The Bible contains plenty of reasons why believers in God should not desire to see an earthly kingdom come to pass by the will of people. The Jewish Testament is all about God's reluctance to see a people have the king they begged for. Jesus made sure that Rome and God were given their separate due.