"All we've got is Irish mythology, usually gathered by Christian monks. So it's basically the same as Germanic mythology, except with a much stronger link to the medieval world and Christianity."
That... seems a bit simplistic. (There've been some arguments that the 'happy ever after' interlude post-ragnarok is a christian insertion into the original norse myth cycles, for example, but I won't pretend to deep expertise on the topic.)
Aside from some broad similarities in culture and theme (fierce warriors flying into battle rages, the gods vanquishing/intermarrying with a race of giants/titans/jotun/fomor, etc., which I'd guess is more of a PIE motif in general) I'd say that norse mythology is distinguished by the funky multi-dimensional nine-worlds cosmology. For me, irish mythology is distinguished by the sense of... deep cyclical history, I suppose? Lebor Gabala Eireann is an account of wave upon wave of invasion washing up on the island's shores for millennia, in ways that I suspect were an archaic echo of real events. (At least two waves of hunter-gatherers, later displaced by neolithic peoples, then displaced by early proto-indo-europeans in turn displaced by their iron-wielding gaelic relations, followed up by vikings, normans, and english planters.)
"You look at Secret of Kells, Song of the Sea, Wolfwalkers or Mage's Bride and the clear connections with the Christian world are always there. Norse we-wuzzery dodged all of that, for some reason."
I'm not sure what you mean here, specifically? I would more-or-less agree with the sparsity of non-Irish sources, and that a lot of the aesthetic motifs that people think of as quintessentially 'celtic' are in fact borrowed from exogenous sources (the interlocking animal knotwork figures are all norse, ogham is derived from latin, etc.) (I love Jim Fitzpatrick's work, but he's terrible for this.)