If you look at the sources for instrumental dance music, there is FAR more English music known from before 1800 (Playford to Rutherford, say) than there is Irish. And it was extremely influential - look at Alois Fleischmann's "Sources of Irish Traditional Music" to see how many English tunes the Irish adopted (a similar story could be told about Scottish music, though we were always the most productive nation of the British Isles for folk music on paper).

God knows what you think the mediaeval church had to do with it. It was entirely irrelevant to secular folk music everywhere in Western Europe, the British Isles included.

Waffly crap about being able to hear distinctions between English and "Celtic" music just make you part of the problem. You don't need your ears attuned, you need better information.