In general, yes. I do not know if the same issue arises when using AF, but on the KL website I frequently see tickets to the US for which the DL domestic segments are as often as not in Economy as in Business. You can take it for granted that the transatlantic leg will be in business class, but dealing with Delta domestically is always hit-or-miss. If you move forward or backward a few days in the calendar view, you might even see the same exact routing for the exact same price for which the domestic sector(s) is/are in Business ("First"), so it might pay to be a bit flexible on dates. If you look a few days before and a few days after your intended travel dates, you might find this ticket at the same price and all in J.

I don't know why this is, and I don't have the same experience at looking up tickets involving other partners, but it seems that with Delta they don't always get access to all available J fares and, rather than not show the itinerary at all, they do show mixed-cabin bookings which are not explicitly flagged as such.

In general, I don't mind sitting in shorthaul Y on a short sector e.g JFK-DCA or the like, particularly if it slashes €500 off the price compared to other itineraries, and even if on other dates I might be lucky and find through the website that I could sit in J on that short sector.

But you do have to watch out. If it's important that every sector is in J - and for me that would really only make sense on a transcontinental, but for trips to the West Coast I would anyways completely avoid booking any itinerary that required a DL transcontinental flight anyway, preferring to take the widebody all the way to the West Coast first - why forego the greater XPs available on West Coast routes for the measly domestic XPs? - but sometimes, with the price differential, I don't mind taking a short DL hop in Y if I can't find J on my chosen dates.