why would a KLM flight not show up on the KLM website?

My guess would be that you are caught up in the intricacies of code sharing between KLM and Delta. One side is making an offer the other won't.

In terms of fare buckets and tickets repricing, sometimes old data gets cached in the flight search system in order to make searches quicker for you and reduce computer load for the airline. So the cached search could be showing ticket types that were already sold to someone else; the system just hasn't quite updated yet. If you're trying to buy more than one seat, the computer rules say that all the fare classes have to be the same (and the typical wide body plane has 20+ different fare classes) so it will give you two tickets that are, say, B type fares because it's the cheapest pair left on the flight, even if there is one V and one T type fare (both of those are deep discount economy fare types) and each of those would be cheaper than the B fare.

The only way to see if the price you're being shown is live or cached is to just go all the way to the purchase page and see what turns up. And remember that you do have 24 hours to cancel a ticket purchase for a full refund when the route touches US soil (US government regulation) so no worries aboyt accidentally clicking the wrong button there.

Delta, Air France/KLM, and I think Alitalia operate transatlantic routes under a joint venture agreement so if you book through Delta, then you're under Delta's fare rules for the ticket purchased there no matter which of those airlines actually fly the route and all will be well. The KLM-marketed ticket may have slightly different fare rules preventing that routing or there may just be a quirk in the computer system that prevents that route, which is not uncommon- it could be if you called KLM's customer service with the flight details, they could manually book the ticket for you.

If the price is right booking through Delta, I'd just book through them online and not worry about it.

I don't think I'd have a problem making the purchase, either. I would definitely check that the reservation was still there for the expected flights pretty frequently, though!

Delta, KLM and Air France have a rather unholy alliance. If you go ahead with the purchase, I would strongly recommend you check fairly frequently that the flights you are counting on don't get changed. If they do, good luck resolving it - spent 6 hours on the phone between the three airlines trying to resolve AF's decision to re book my husband's ticket to a day prior to our original departure - which did not get resolved to our satisfaction and resulted in us being put on a Delta flight (4 additional hours of travel, thank you) and in a lower class - for of course the same price.
Ah, the joys of modern travel.....

We had a similar, but reverse thing once with British Air and American. We booked Tulsa - DFW - Heathrow, The TUL - DFW was codeshare with American. We were informed by BA that that leg had been canceled, and they rebooked us on a much less convenient AA flight to Dallas. I checked with American, and the original Tulsa - DFW flight was still listed. I called BA and they insisted it was no longer available, that their system showed the flight no longer existed. It continued to show up on AA for weeks, until I quit looking.

Had it been closer to booking time, I would have canceled the first leg and just booked the flight from DFW to London, and taken the convenient AA flight to DFW. But I wienied out; what if it really did get cancelled, and I'm stuck paying full fare one way?

So I guess what got canceled was the code share.

Delta, KLM and Air France have a rather unholy alliance.

Nancy, how is the SkyTeam global alliance you mention above any more "unholy" than Star Alliance or OneWorld?

While I can't explain why the flights appear differently on different websites, I would feel comfortable booking the entire trip through the Delta website.

Yes, there is a chance the flights may change between now and then. That's just the downside of booking your flight early. It has happened to me on almost every itinerary I've flown. Something always changes. Just check your flights on a regular basis to see if there's a time change or seat change.

And if you haven't signed up for an airline's frequent flyer program, go ahead and do it now! In an IRROPS situation when flights are cancelled or delayed, it may not be a golden ticket but it could put you ahead of other flyers who don't have a ff# with said airline.

KLM is owned by Air France and Air France and Delta have agreed that for flights originating in the US, Delta names the price and Air France will not undercut it.

Unholy because it allows them to circumvent the EU rules regarding flight cancellations and delays.

Go to Google Flights and try to see if the flight is available. I don't know the date you plan to travel but I tried a few different days and there were no flights that arrived at 1:30 PM.

I also suggest you clear the cookies on your browser and try again.

Unholy because it allows them to circumvent the EU rules regarding
flight cancellations and delays.

This remains unclear. What is "it" to which you refer?

Whatever that definition is, only SkyTeam circumvents EU rules but OneWorld and Star Alliance don't? What proof do you have that SkyTeam circumvents EU rules? Nothing below about Delta or Skyteam circumventing EU laws. Please help.

https://thepointsguy.com/2014/02/how-to-get-eu-passenger-rights-compensation-from-us-carriers-for-delayed-flights/

https://www.claimflights.com/airlines/delta-air-lines

https://www.euclaim.co.uk/flight-delay-compensation/delta-airlines

https://www.delta.com/content/www/en_US/traveling-with-us/ticket-changes-refunds/exit-eu-compensation.html

Proof? How about re booking my husband's ticket to a day prior to our original booking and then subsequently lying about rationale for ticket change by Air France? So how do I know they were lying? Because I had booked the same return flights (with a reserved seat adjacent to my husband) but under a completely different reservation since my husband was flying to Europe to meet me.
The lies got even more interesting when we were turned over to Delta since Air France refused to honor his ticket. Air France had informed my husband that he could only fly on Saturday OR the following Monday - that there was not an AF flight available on Sunday. Thus, since he had to be back for a meeting on Monday, we had no choice but to work with Delta to get a flight on Sunday. The Delta rep "confirmed" the AF story - Interestingly enough, I not only did not receive notice of this flight "cancellation" but I also received notification from Air France to "check in" for my flight.!!! Delta then came up with the excuse that the flight was "delayed" and thus since he'd miss the connecting flight, had had to be re booked. When I asked why I hadn't been rebooked, since I was on the same flights, the Delta agent had the nerve to suggest that they contacted me in the same email. When I informed the agent that that was impossible since my husband and I had different reservations with different email addresses, the agent said that was not the case - seriously?
KLM was completely unresponsive - after trying for4 hours to reach them, we simply gave up. And why did we contact KLM? Because the Delta rep said that although we'd paid for premium economy on AF, Delta couldn't accommodate the request - THAT had to be handled by KLM - which of course, was not available when we called, repeatedly.

Each airline "pointed the finger " at the other and each denied any responsibility. By re booking my husband to a flight a day earlier, AF was able to evade the fines that the EU imposes when passengers are bumped.
Thus, my reference to an "unholy" alliance.

Today (24+ hours later) all the flights are showing up on KLM. In fact, the itinerary I wanted yesterday was the first one to pop up today. Albeit for the more expensive price (the price the Delta agent quoted me on the phone. So maybe the flight I was looking for disappeared and then was re-posted on KLM at a higher price? In any event, it's no longer my best option....so I'm still watching and waiting. Any value in waiting until the very end of the month hoping the fares might fall? (I'm going to need a vacation just from planning this part of my vacation!)

So sounds like a computer system issue. All the airlines will have those times when they're slightly restricting what you can book and change because they're working on something on the back end (I know Delta can get weird with seat assignments and such on Saturdays when they're tweaking their schedule and might need to swap a plane assignment for another plane with a slightly different seating chart)

They may reprice the fare buckets that are left but there are no guarantees. If you've got a little schedule flexibility, moving the dates by a day or two can sometimes find you some of the tickets in the cheap fare buckets again.