The mainstream of religion largely does throw away inconvenient facts from science.
Compromising science to appease religion is a seriously bad idea (I know that few specifically argue for this, though some do, but many proposals do this in practice)
Even religions that are not explicitly anti-science usually only give lip service to science (for example the catholic church's silly standards for miracles)
All of these ideas have been played out before, but my own contribution is this thought: in the modern media cycle interesting debates and ideas are much more powerful than bland ones. People live for arguments, it's the entire reason for existence of reality tv. This is not always a good thing (the entire history of the Republican party after Nixon is based on the idea of interesting but bad ideas), but it is a fact. An exciting debate between science and religion is going to get a lot more attention, and probably people interested in science, than mealy-mouthed accommodation. Now, this will certainly lead to a lot of people coming down against science, but I view it as similar to the story of the civil rights movement (and gay rights movement): there was a lot of abstract argument about equality and justice, but there was very little progress in the public realm until there were those who were unafraid of being called strident or militant (and came out of the closet).
Now, science outreach is a very different thing from the (gay) civil rights, so I could be totally wrong about this, there probably is no research on it. But I do tend to think that it's impossible to make much progress on an issue without a strong viewpoint. People may be ignorant, but they are smart enough to tell when you are pandering to them, and they don't like it.