It's not true, but in discovering that I discovered something more interesting.
The solar azimuth at dawn on the Winter Solstice in our age is 128° (to the nearest degree). However, Sirius's dawn azimuth is 116°.
Now, Sirius's dawn azimuth doesn't change much over time, it's directly related to its celestial declination (effectively its 'latitude' on the celestial sphere) which is - while not fixed - stable over long periods of time.
However, the Sun's declination does change - moving from +23.5° to -23.5° over the course of a year - being at its lowest at the Winter Solstice.
That means there must be a day in the year when Sirius rises before the Sun and the Sun's declination is equal to Sirius's, meaning that their dawn azimuth will be the same and Sirius will have indicated the impending dawn's sunrise position exactly.
The statement I encountered mentioned that the effect of seeing Sirius rise where the Sun would rise at the Winter Solstice was more pronounced around 2000 years ago, and used this point to link into the story of the Three Wise Men (the belt stars of Orion) following a star (Sirius) to locate the Son of God (Sun).
I decided to check if I'd overlooked something by seeing if the alignment does occur around Winter Solstice 1AD - it doesn't. Sirius's dawn azimuth is 115° while the Sun's remains 128°
But, realising that the Sun would have a declination equal to Sirius's at some point earlier in the year (this happens twice, actually, but only once when Sirius rises before the Sun), I tracked back and found when it would be.
It's on November 8th 1AD, or Samhain (one of the cross-quarter days in the pagan/Celtic calendar).
I thought I was content with that. On Samhain, Sirius preceeded by Orion's belt stars indicates the forthcoming rising position of the Sun quite precisely.
However, I wasn't prepared to find that there was the "Star of Bethlehem" (a
triple quadruple conjunction of Venus, Jupiter, Mercury and Mars) rising immediately before the Sun on the same day.
Samhain 1AD Dawn and the Star of Bethlehem
Updated: That'll teach me to zoom in! In the original version of this article I'd only spotted the triple conjunction of Venus, Jupiter and Mercury. It was simply that I'd not zoomed in far enough to see that Mars was also part of the conjunction, making it quadruple, not triple.