wooden way

Michael Sandelの "What Money Can't Buy"、ギフトの話はまだ続いています。
A man contemplating a gift for his girlfriend "has private information that the girlfriend would like to know: Does he really love her? Choosing a good gift for her is a signal of his love." Since it takes time and effort to look for gift, choosing an apt one is a way for him "to convey the private information of his love for her," This is a strangely wooden way to think about lovers and gifts.
私が気になるのはギフトの選び方ではなく、"wooden way" の意味です。
・Oxford English Dictionary: Stiff and awkward in movement or manner: she is one of the most wooden actresses of all time
・Merriam-Webster: lacking ease or flexibility: awkwardly stiff: a wooden speech: a wooden performer
・Cambridge Dictionary: used to describe behaviour that is awkward or shows little expression: She gave a wooden smile to the camera.: I thought the lead actor gave rather a wooden performance
意味はちょっと違いますが、日本語の「木で鼻をくくる」を連想しました。