You are right to observe that the play seems short on celebrating the nativity and long on slapstick humor, but this play is considered perhaps the best and most representative of the plays in what is known as the Miracle/Mystery Cycle and is certainly the most successful of The Wakefield Master's plays.
Divided into two parts, a long first part on a sheep-stealing episode that is a parody of the nativity and a shorter, more serious, nativity scene. As as is usual with The Wakefield Master, the initial dialogue among the shepherds is a criticism of their lot in life:
We ar so hamyd,/ Fortaxed and ramyd,/We are mayde handtamyd/Wyt these gentlery-men.
In other words, the common plight of the common man is to be hamstrung, over-taxed and beaten down, and made submissive by the upper class. The use of the shepherd's allows the playwright to identify ways in which the lower classes are abused, and this sets up their joy at the actual birth of Christ, with hopes of a better life.
Because the sheep-stealing scene by Mak and his wife, Gil, is meant as the major entertainment of the medieval audience, which consisted of a mix of the peasant and yeoman classes, it is long, complicated, very funny, and is meant to provide a parody of the nativity. After stealing the sheep, Mak and Gil disguise it as a baby and put in a cradle. When suspected by the shepherds of stealing their sheep, Mak invites them to his house, and Gil is standing by the cradle when she says:
I pray to God so milde,/If ever I you beguild,/That I ete this childe/That ligys in thi creydll.
Of course, after a series of slapstick attempts to see the "child," the sheep is discovered in the cradle, and that resolution moves the play to its serious subject, the actual nativity scene, which is only two or three pages of the entire play.
One of the hallmarks of plays by The Wakefield Master, which has dramatic implications for later drama, is his introduction of common people, such as the shepherds, into drama, and from that point onward, the common man, along with his troubles with the rich, become a staple of drama.