When you're regularly calling everybody else by their name, calling yourself by your name makes sense.
However, do keep in mind this is anecdotal so I don't know how much it applies to more broad settings.
Actually, calling the other one by their name + san/chan/kun is the most common way. Using "anata" (= you) can even be considered rude under certain circumstances. My wife never calls me "anata". Also, pronouns are often omitted if it is clear who is being meant (a bit like in Italian).
Using or not using the pronoun has the same emphasis as saying to someone:
'You're drunk.' versus pointedly saying 'You, are drunk'.
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Soy Latte, por favor! (In Spanish this translates literally as 'I'm a latte. please.')
I'm not a linguist, but it seems like there's no grammatical difference between first/second/third person in Japanese - you can consider all the sentences to be in third person.
One can imagine growing up as a child and hearing Japanese with its variety of first-person pronouns, which also are not used as much as we use them in English, and choosing to use your name as it is consistently associated with you.
That said, I've had only a smattering of Japanese study, so definitely take all of that with a grain of salt.