I don't think people who are giving their charity to churches are trying to give to the most efficient charity.

This is sort of the point. The more one is concerned with efficiency (like those who are concerned about the marginal benefit of a great charity vs. the best charity), the less an info-lag will have an effect on him/her. Effective Altruists who are thinking at the level of "I should only give to one absolutely best charity, to maximize marginal value", are not likely to pick a charity and give to it blindly, never again looking at the least bit of evidence of marginal effectiveness.

I'm willing to wager that the marginal value lost due to an info lag when everyone is giving to only the best charity and aren't paying enough attention, is vastly outweighed by the value produced due to everyone giving to the absolute best charities.