Why is Gaedheal spelt with an 'e' after the 'dh' if the diphthong 'ae' is broad?

I guess it shouldn't be, by today's rules. But then, we eliminate the "dh", too, so I've never seen "Gaedhal" as a spelling, only "Gael". In the old days, we had "Gaedhilge", as in "Irisleabhar na Gaedhilge", a similar violation of today's rule. I suppose these earlier spellings made sense at the time, given that the form that came before "Gaedheal" historically was Old Irish "Goídel", in which the "oí" was a slender diphthong.

I have a strong idea, but I can't be sure, that the same 'ae' diphthong, is what you might called a "wild card" in grammar.It can be either slender or broad depending on context.

Níl sé sin fíor anois. Bíonn "ae" leathan i gcónaí i nGaeilge an lae inniu. Féach, mar shampla, nach scríobhtar "Gaelge". Tá an "i" riachtanach mar tá "lg" caol. Agus an difríocht idir "Gael" (uatha) agus "Gaeil" (iolra) - bíodh nach bhfuil cuid de na Gaeilgeoirí in ann an dá "l" sin a idirdhealú ina gcuid cainte.