Actually, that is not quite right. A House representative requested specific documents already gathered by the State Department. The delay you mention would only happen if the State Department refused to comply with the Senate subpoenas. Of course, Trump ordered every person in his Administration to defy the House subpoenas, and he probably would have done the same to Senate subpoenas. But Republican Senators chose to represent their own party instead of their own branch of government.
Here's the rub. Republicans criticize the House for not waiting for long court battles to obtain subpoenaed evidence and then support Republicans in the Senate after they voted to issue no subpoenas, not even one. We are witnessing partisan corruption by Republicans which most voters will not approve.