That would be the same Cockhill Celtic who back in September lost to Enniskillen Rangers in the Presidents Cup Final, a trophy contested between the IFA and FAI Junior Cup Champions?
https://www.fermanaghandwestern.com/...tail/10092196/
For information, Ekn Rgrs are one of five football clubs in a town of 14k people, which incidentally has one GAA club and one Rugby club. I'd venture that their (self-owned) ground is better than that of many (most? all?) of those Intermediate clubs you mention, yet they are a resolutely Junior club in NI terms.
The point being that as the pyramid in NI has developed over the last couple of decades, it has done so by first spreading out from the existing centres of football - Belfast, Antrim, Mid-Ulster etc - then into newer, less traditional areas, to incorporate the likes of Dungannon Swifts, Ballinamallard Utd and Warrenpoint Tn.
I have no doubt that had the IFA attempted to parachute such clubs into the upper tiers of IL from the start, rather than requiring them to work their way up to Senior status as and when they merited it, then they'd have died on their arse long before they got there.
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