Yuo only need to sever one or two of the nerves for this to happen. That was a nice picture of face muscles your provided, but has little to do with the nerves. Most of these muscles and their nerves lie on the outside of the skull. A deep cut to the nerves could still be true.
"the facial nerve" is a misnomer, there are too many.
Branch of CN VII Location of Lesion Actions
Posterior auricular Posterior auricular Pulls ear backward
Occipitofrontalis, occipital belly Moves scalp backward
Temporal Anterior auricular Pulls ear forward
Superior auricular Raises ear
Occipitofrontalis, occipital belly Moves scalp forward
Corrugator supercilii Pulls eyebrow medially and downward
Procerus Pulls medial eyebrow downward
Temporal and zygomatic Orbicularis oculi Closes eyelids and contracts skin around eye
Zygomatic and buccal Zygomaticus major Elevates corners of mouth
Buccal Zygomaticus minor Elevates upper lip
Levator labii superioris Elevates upper lip and midportion nasolabial fold
Levator labii superioris alaeque nasi Elevates medial nasolabial fold and nasal ala
Risorius Aids smile with lateral pull
Buccinator Pulls corner of mouth backward and compresses cheek
Levator anguli oris Pulls angles of mouth upward and toward midline
Orbicularis Closes and compresses lips
Nasalis, dilator naris Flares nostrils
Nasalis, compressor naris Compresses nostrils
Buccal and marginal mandibular Depressor anguli oris Pulls corner of mouth downward
Depressor labii inferioris Pulls lower lip downward
Marginal mandibular Mentalis Pulls skin of chin upward
Cervical Platysma Pulls down corners of mouth
The pictures I look at show many of these nerves close to the surface, and that lesions will stop them from working, not deep and unreachable by cuts.
I just don't see it as unlikely I guess