The sole question is whether the operating carrier, KL, properly denied boarding. If the ticket was not properly issued in the KL system as the result of a failure unrelated to KL, then there is no compensation due (as EC 261/2004 applies only to the operating, not ticketing, carrier). Thus, what DL did or did not do is irrelevant.
You may file a claim with KL and understand that on the facts you have provided, that it will almost certainly be denied and that the reason for the denial is most likely correct.
Alternatively you could file a customer service complaint with DL. You are not due compensation, but DL might well issue a customer service gesture of some miles or a small credit.
Your instincts were initially correct. Anytime you are unable to check in online, it is essential to call, Don't give up. Put the phone on speaker + mute and multitask, but don't give up. With a day to fix, this was fixable.