Outside the US, how is a month+day combination written out/spoken?

"May 3" / "May 3rd" or "3 May" ?

3 May in Italy. It’s always day first. You never hear anything saying or writing the month first.

In most contexts I would never use the cardinal number "3", it'd always be "the 3rd of May" or "May (the) 3rd", but in a country like Australia (with a high number of speakers from different cultural backgrounds) you hear all sorts of conventions used in casual conversation.

"3 May" in French (« le trois mai »)

“3rd of may”

While I am a strong supporter of ISO8601 (e.g. 2023-07-12) there is an argument that decreasing specificity is preferable (as in ISO8601) and that for most day-to-day usage the year is somewhat implicit, hence being at the end.

Personally, I write the month out if I'm not using 8601 formatting to avoid any ambiguity.

For any sort of recordkeeping, though, I think it's preferable to go full on YYYY-MM-DD. It's more thorough, precise, and sorts properly on a computer.

I don't think "it makes more sense" is exactly describing the reasoning.

Neither do I, but the rest of what I wrote does.