>>>It is entirely true that 'mainstream churches' can have it both ways. Of course, they are political entities disguised in a moral shell. They don't have a problem talking with two tongues at the same time. They say, "The Bible is true and it is the foundation of our faith", and "The Bible is only a mythological picture book and cannot be taken seriously". If in doubt, the mainstream church favors tradition over the Bible, and it favors political power over tradition.
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>This often is said, but what is the basis? Apart from allegations from detractors, specifically where is this to be encountered in 2014?

One good example would be the protestant church (mainstream, taking into account that there are many sub divisions and splinter groups), which bases its viewpoint on Luther's literal interpretation of the Bible, and the claim "Sola Scriptura", and at the same time moving during the last century towards the theory of "Theistic evolution". If you interpret the Bible literally, and use "Only the Bible" as foundation for your faith, how can you declare that God uses evolution to create the flora and fauna, while there is not one syllable that even remotely hints to that thought. If the Bible clearly contradicts and omits this truth, how can you take the rest of the Bible seriously? And at the same time criticize the Catholic church that they do not accept the Bible as infallible and to be their pillar of faith. (At least the Catholic church admits that).
Now, in Europe, all members of the protestant church do pay a fixed amount to the main church which is automatically deducted from their salary (exactly 10% of the amount they pay in taxes to resemble the old testament practice of the tithes), unless they leave the church. (The Catholic church does the same). Also the protestant church has a lot of interest in the school system and in order to be taken seriously they make concessions to their faith. It's partially about money, and partially about power.
(Hitler was one of the more predominant followers of theistic evolution, and he happily read the works of Blavatsky. He just wanted to speed up that natural process and thought to gain favor of God by doing so. In fact he did believe doing God's work on earth and was unpleasantly surprised that those less developed races from the east and west destroyed his dream of 1000 years reign of a purely Aryan nation.)