Being French, I feel pretty well informed when I say that not a whole lot of French folks want to eat horse meat, and the numbers are dropping over time.

https://www.ouest-france.fr/economie/agroalimentaire/la-vian... and other articles appear to agree with me.

I know they do, but is it as popular and as much a part of the culture?

Well the French invented chivalry (=horseman) didn't they? So I would say it was an even stronger part of their culture back then. But I don't know about now.