I think that “Ce n’est pourtant pas faute de + verbe” is a french idiom and it is up to the reader to work out what it is about however there will have a linking word related to the previous sentence for instance s’ afficher

for

Ce n’est pourtant pas faute de s’afficher publiquement avec sa femme
Brigitte.

The clues are “sa femme Brigitte” and “président de la République” in the previous sentence.

For:

Ce n’est pourtant pas faute de nous décarcasser

You need to be consitent with the what you want to recall from the previous sentence and in this case it would be “me décarcasser”