> Because leap day is at the end of the year, but at some time we moved the start of the year two months back (same reason why September through December now are the ninth through twelfth month of the year, not the seventh through tenth)
There is actually no hard evidence that the Romans ever started their calendar in March instead of January. The earliest contemporaneous use of the calendar relies on January 1 as the start of the year, short February, and with intercalation happening after (or maybe within) February.
The primary evidence we use to indicate that the start of the year shifted is... the apparently wrong month names. Some writers did describe a calendar that starts at March and ends in December, with winter basically having no proper calendar--but these are writers describing how their calendar worked several centuries ago, attributing it to mythological figures, and the explanation strikes me as very heavily a "just so" explanation.
If you want my hypothesis, the Roman civil year never started in March. But March would have had some amount of primacy, as it indicates the start of the planting year. Shenanigans in the calendar would have occurred in February to ensure that the equinox is properly timed to happen in March. But the civil year would have started closer to the beginning of winter for other reasons (perhaps taxation? but finding this level of granularity of information on Roman taxation is difficult).
In this hypothesis, the month names were not incorrect because they were never intended to count from the beginning of the year. Note that the first 6 months of the civil year have names, while the last 6 are merely numbered. It makes no sense to me that you'd make up special names for the first 4 and the last 2 months of the year, while skipping everything in the middle.