The Julian calendar is, it seems, based on the Roman calendar. The oldest Roman calendar began in March (https://www.timeanddate.com/calendar/roman-calendar.html) and may have been tied to the equinox. And as you found, the Julian calendar (when instituted by edict by Caesar) started on January 1 (https://en.wikipedia.org/wiki/Julian_calendar)

I don't see any tie to Sirius.

There would be no seasonal impact with Sirius (as there is with the Nile Floods) for any place outside the Nile River valley. Nor did they need astronomy as by then the had a fairly precise measure of the length of the year. Caesar was using Greek astronomers and mathematicians and neither of these had a focus on Sirius that I'm aware of. The Greek calendar certainly didn't use it.

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