Tagaim ar is inappropriate here. It should just be Tagaim an chéad lá d'Eanáir. None of the sample sentences in the EID page for "come on" use ar when there's a temporal meaning. As an example "He's coming on Sunday" is given as Tiocfaidh sé Dé Domhnaigh with just the plain verb tar. "To come on alternate days" is given similarly, teacht gach dara lá. There are also some sentences from the FGB that support my hypothesis. Tháinig sé an lá corr cointinneach and Féach an lá a tháinig sé are two examples. The use of the direct relative in the last example is particularly telling. None of the meanings on the FGB page for tar ar include "arrive on" in a temporal sense, only a physical one. This is a good reminder that verbs that are phrasal in English aren't necessarily so in Irish and vice-versa.
Dé Domhnaigh is already adverbial - the "on" in "on Sunday" is contained in it. That's not the case with an chéad lá d'Eanáir.
Translating "To come on alternate days" as teacht gach dara lá is all very well, but it doesn't support your argument because "to come every second day" doesn't use "on" either. "See the day that he came" has the same issue. You might have a case with Tháinig sé an lá corr cointinneach, but it's a sufficiently strange sentence, with the suggested translation of "he picked a most awkward day to come", that I wouldn't want to rely on it for guidance.
You're correct that the entry for tar ar doesn't give any temporal examples, but that doesn't actually exclude such usage.
The EID, on the other hand, has a number of temporal examples:
"He arrived on the dot" - tháinig sé ar an bpointe
"To arrive on the stroke (of time)" - teacht ar an spriocuair
"He will arrive on the 10th, when he will open the new school" - tiocfaidh sé ar an 10ú lá, agus osclóidh sé an scoil nua
The NEID also has a number of clear examples of tar ar being used temporally:
"she's coming on the first" - tá sí ag teacht ar an gcéad lá
"it arrived on the promised date" - tháinig sé ar an dáta a bhí geallta
Dé Domhnaigh may function adverbially but grammatically it’s a noun. A similar idiom is teacht an athuair where teacht is directly followed by a noun acting as a temporal modifier, similar to tágann sé an chead lá would be in this sentence. The whole phrase is acting adverbially.
That the corresponding English can be phrased for teacht gach dara lá and féach an uair a thaining sé also prove my point perfectly well, teacht is directly modified by a noun without any intervening preposition. That they can be similarly phrased without the preposition in English is of no consequence to the Irish phrasing. The construction of sentences in English isn’t the determiner of their phrasing in Irish. This shouldn’t be surprising, since they’re two different languages. It’s also very convenient how you dismiss out of hand Tháinig sé an lá corr cointinneach because you find it “strange.” The dictionary gives a loose translation sure, but the sentence itself is incredibly straightforward. There’s no verb for “pick” anywhere in there, just the verb for “come” and information saying when he came. It’s only “strange” if you demand these two elements be connected by a preposition, a demand ultimately rooted in the logic of English.
I won’t play the fine-toothed comb game where we try to nitpick each other’s sentences. Instead I’ll add this quote I found in some old commentary on the matter,
He goes on to give another example and counterexample: “Bhí sneachta ann an tríú lá den mhí but bhí troscadh ar an Satharn so a ghabh tharainn.” The sentence here is clearly narrating the occurrence of a repeated action, so ar shouldn’t be used.
I note that you have modernized the good Father's spelling, so you acknowledge that Irish wasn't pickled in aspic in 1926, and that the language can change over time.
What I found strange about the Tháinig sé an lá corr cointinneach is not the loose translation, but that the implication of frustration and indignation in the suggested translation indicates that there is a lot more going on in that sentence than meets the eye. That makes it a less than ideal model for other circumstances.
I agree this example sentence implies a certain frustration and possibly has interjectional character but this implication doesn't invalidate it as a model. Grammatically it's a very simple sentence, so unless you think mild exasperation causes Irish grammar to completely change there's no reason to suspect it's structurally atypical. This is especially true given it's the only example of the form Tagaim an lá... with or without ar in the whole FGB. In light of Ua Laoghaire's comments it seems quite a strong example actually.
Your remark about spelling is entirely specious. How languages are spelled relates little to how they're spoken. The English writing system could switch to cuneiform tomorrow and it wouldn't matter one bit the spoken language. There's virtue in having spelling that more closely resembles how words are pronounced, which is why I chose to modernise it. This choice doesn't impugn the quality or the relevance of the Irish contained therein at all.
Languages do change over time, but native speakers are usually in the driver's seat. Irish is one of the only languages where the vast majority of "speakers" didn't grow up with the language and don't actually speak it regularly. In light of this equating the changes wrought upon Irish with the changes in other languages would be a mistake. That's why Ua Laoghaire and other Gaeltacht authors are still the gold standard.
And yet the entry for ar on it's own includes Ar an lá deireanach - "on the last day". Even the EID allows for ar in it's "expression of time" section of the entry for "on" : "On May the third" - (ar) an tríú lá de Bhealtaine.
I'm quite happy to acknowledge that this is an incursion by the preposition ar that people were objecting to a hundred years ago. But it is now a part of the language as it is used today by many Irish speakers.
Ar can certainly be used temporally in some contexts e.g. ar maidin. This is something I've never objected to. If that was unclear I apologise. My specific objection is to the use of ar in this example, a usage not found in Ó Dónaill and one specifically condemned by traditional sources. If you choose not to take heed of that my only reply is beatha dhuine a thoil.
If the 1 in your example is meant to indicate the first day of January, we don't use a comma in English. The comma separates a month or a month & day from a year, not a month from a day within it: January 1 (or January 1st, or 1 January, depending on where you are) vs. January, 2020 or January 1, 2020.
If you typed "January, 1" and Duo didn't accept your answer, maybe try it without the comma? Duo might have interpreted the 1 as a year instead of a day.
Numbers like "first"/céad, "second"/dara, "third"/tríú are "ordinal numbers" or orduimhreacha. They aren't covered in the course notes, but you can read a bit more about them at Gramadach na Gaeilge.