Except for the historical references to the rabbis and Josephus.
I'd have to look up the Talmudic reference, but to be blunt about it, Edersheim is ignoring basic geography: … Sorry, it still doesn't add up. Nice try, though.
As I understand it Edersheim is essentially conveying the historical reality of that time wrt the flocks used to the sacrifices. While I don't necessarily accept the December timeframe in any authoritative way, I do find the typological suggestion of the shepherds coming to attend the true Lamb of God fascinating.
But recall Edersheim’s response in the footnote to this alleged issue:
954 The mean of 22 seasons in Jerusalem amounted to 4.718 inches in December, 5.479 in January, and 5.207 in February (see a very interesting paper by Dr. Chaplin in Quart. Stat. of Pal. Explor. Fund, January, 1883). For 1876-77 we have these startling figures: mean for December, .490; for January, 1.595; for February, 8.750 - and, similarly, in other years. And so we read: ‘Good the year in which Tebheth (December) is without rain’ (Taan. 6 b). Those who have copied Lightfoot’s quotations about the flocks not lying out during the winter months ought, at least, to have known that the reference in the Talmudic passages is expressly to the flocks which pastured in ‘the wilderness’ ({hebrew}). But even so, the statement, as so many others of the kind, is not accurate. For, in the Talmud two opinions are expressed. According to one, the ‘Midbariyoth,’ or ‘animals of the wilderness,’ are those which go to the open at the Passovertime, and return at the first rains (about November); while, on the other hand, Rabbi maintains, and, as it seems, more authoritatively, that the wilderness-flocks remain in the open alike in the hottest days and in the rainy season - i.e. all the year round (Bezah 40 a). Comp. also Tosephta Bezah iv. 6.
Which refer to a time when Jerusalem had been under siege since Passover, many had died because of famine and plague, and the services had been disrupted for lack of animals to sacrifice. There's another Talmudic reference which tells us that Aviyah served just before Shavuot (Pentecost), as I recall. I'll have to look it up when I get home.
But recall Edersheim’s response in the footnote to this alleged issue
Which presents two differing rabbinic opinions. We could favor one over the other, or figure out if there is a way to reconcile them. (Such as supposing that there were two traditions based on there being practices in different parts of Judea, depending on the local geography.) Either way, since there is nothing conclusive, and nothing with deals with the geographic realities in Bethlehem, you are left with only an opinion that doesn't quite fit with the facts, as other commentators have recognized.
Shalom.