I would say that 愛の営みをする is indeed literary speech. The feel is similar to "to make love", which is certainly the best translation. I understand it to be neither particularly passionate, nor perfunctory, but factual. The phrase can be changed in the following way, making it more and more suited for conversation:
- 愛の営み (practically not used in conversation)
- あっちの方 (sometimes used in conversation)
The top of the list would be more likely to come up in a conversation with your grandma, who wants to understand the chances of her seeing her great-grandchildren. The bottom might come up in a conversation with your boss, seeing that you have been acting discouraged for a while and enquiring how things are going between you and your wife.
In conversation amongst younger people, 寝る "to sleep (with)" or やる "to do it" are more common, just like in English, I would say.