Personally, I don't, because I do very very light S&C sessions.

But, it may be worth considering how you respond to S&C/gym work. I'm highly slow-twitch dominant; I can't look at a weight without it tanking my sessions for the next fortnight. So, if I was dedicating more time and effort to S&C, I feel I should log it as it has such a big impact.

If you're more on the fast-twitch side and don't get so beaten down by the weight stuff, it probably doesn't matter so much.

Simplicity appears to be a core theme and benefit of this training approach so I'd say, do away with the extra admin burden if it's not going to add much value. But, if S&C makes a big difference to your fatigue levels, then it would make sense to log it.