I've started billing that. Not always, rarely in full, but I bill some of it. It's not just for the money but started more like "what exactly do I work? Best way to know is to bill it".
Unless you're a contractor, exempt salaried employees are exempt from overtime requirements and thus typically can't bill for overtime, unless their employer likes burning money for some reason.
It goes both ways though - if I can't stop thinking about something that's just not quite working yet, then I haven't/can't 'clock out'.
I think I'd like a job with truly flexible hours: work all evening on something if it's occupying my thoughts anyway, but then get a full 'evening' to chill, full night's sleep, and start the next day when I'm ready. I can do the former, but I'm still going to be expected to dial in ~to pointlessly summarise it~ for stand-up in the morning. (No I don't have young children, yes yes this wouldn't work for everybody.)
So you don't really stop working at 6pm..