Antiphon/psalm/antiphon/psalm-prayer is the proper way. When the psalm is chanted, the last note of the psalm is meant to gracefully take you to or near the first note of the antiphon, so to put the psalm-prayer first would be dissonant. The antiphon is also usually a verse from the psalm or seasonally-appropriate scripture verse or paraphrase, so it makes more sense to include the antiphon as part of the psalmody before and after. The psalm-prayers are an optional extra and are a reflection on the psalm, not directly from the psalm or bible itself as the antiphons are.
FWIW, they aren’t even in the French LOTH, nor even the Latin editio typica! I have them in French in my monastic breviary though.
And to end any “it was ICEL nonsense”, it wasn’t. They came from a Spanish Benedictine monk. But they didn’t catch on. Even though they were in our abbey’s breviary, they never used them. When they reissued new books of their own rather than ones published elsewhere, they didn’t include them.