I've had to time to have a better look at this. The Tusser poem is:

DESCRIPTION OF THE FEAST OF THE BIRTH OF CHRIST (COMMONLY CALLED CHRISTMAS)
(Thomas Tusser)

Of Christ cometh Christmas, the name with the feast,
A time full of joy, to the greatest and least;
At Christmas was Christ, our Saviour, born,-
The world through sin altogether forlorn.

At Christmas the days do begin to take length,
Of Christ doth religion, chiefly take strength:
As Christmas is only a figure or trope,
So only in (c)Christ is the strength of our hope.

At Christmas we banquet, the rich with the poor,
Who then, but the miser, but openeth his door?
At Christmas, of Christ many carols we sing,
And give many gifts, in the joy of that king.

At Christmas, in Christ we rejoice, and be glad,
As onely of whom our comfort is had:
At Christmas, we joy altogether with mirth,
For his sake, that joyed us all with his birth.


VARIATION (c) is Christ the strength &c

Source: Thomas Tusser: Five Hundred Points of Good Husbandry (1812 edition, ed William Mavor)



Tusser was born between 1515 and 1524. He wrote One Hundred Points of Good Husbandry in 1557, which in 1573 was expanded to Five Hundred Points of Good Husbandry. He died in 1580. As far as I can see, this was not included in One Hundred Points of Good Husbandry.

It did appear in a few other books at the start of the 19th century:
Douce -Illustrations of Shakespeare and of ancient manners (1807) and
Southey -Selected works of the British poets from Chaucer to Jonson (1831).

The quote many carols we sing appears in books up to recent times.

My guess would be that someone either misremembered the Tusser poem and mixed up the lines, or that someone took it as the basis for a carol. Searching doesn't give any other references for many carols we sing. There is also the possibility that Tusser was influenced by an existing carol and took his words from that, but nothing else shows up in search that would make that likely. (The awkward endings of the first two lines in the collected version look like misremembering to me,)

Mick