AFAIK, proper nouns are lenited in the genitive by default when modifying other nouns (e.g. teach Sheáin but not feiceáil Sheáin; not entirely sure about the latter), but the Irish Government Educational Council for Gaeltacht and Irish-Medium Schools is apparently called An Chomhairle um Oideachas Gaeltachta agus Gaelscolaíochta, where both Gaeltachta and Gaelscolaíochta are unlenited. I'm guessing the rule only applies to certain types of proper nouns or something?

Gaeltacht and Gaelscolaíocht are not proper nouns, afaik...

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Really? Gaeltacht is always capitalized here.

Capitalization is just an orthographic convention. It doesn't mean anything.
Here it is due to the "Gael" part. Words with ethnic meaning are capitalized.

I'd define a proper noun as definite by default, naming a unique entity.
But "Gaeltacht" just means "a territory where Irish is spoken".
There is more than one Gaeltacht and all of those Gaeltachts have names on their own (Gaoth Dobhair, An Rinn, Cois Fharraige, etc.)

You can make it definite with the article and demonstrative words: an Ghaeltacht seo (this territory where Irish is spoken)

I thought the very fact that it is always capitalized means that it is a proper noun, but even if you grant that, I see what you mean; Gaeltacht is not definite. While I'm here, is it feiceáil Seáin or Sheáin? I got way more results for labhairt Gaeilge than labhairt Ghaeilge, but I'd just like to make sure.

This is a confusing convention in Ireland, even in English language papers you'll see things like:

"a Gaeltacht area"

"I went to a Gaeltacht last year"

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The dialect I use is Munster Irish, particularly Cork Irish, so words or phrases I use might not be correct for other areas.

Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

Alright, but is it feiceáil Seáin or Sheáin (or S(h)eán)?

Sheáin leis an H.

So I guess it really is labhairt Ghaeilge?

I would guess (but that’s only guess of an at best intermediate learner) that a good rule of thumb for what is and what is not a proper noun in the meaning of ‘a word that lenites in genitive in Irish’ is whether the noun may take a definite article or not.

AFAIK proper proper nouns never take definite article, you won’t say *an Seán nor *an Sheáin/*an tSeáin, you will only say Seán, Sheáin. As you won’t say *an Cháit, *na Cháit/*na Cáit, you will only say Cáit, Cháit.

But you can say an Ghaeilge and na Gaeilge, and thus gramatically it is not a proper noun and without the definite article you also will say just Gaeilge and Gaeilge, without lenition.

But sure, for me too the real grammatical distinction between proper nouns and all the other nouns is a bit unclear.