I've never heard about this failure mode before.
"Luckily", there is some logic to it: the BBD chips are configured into the various FX topologies using analogue switches and VCAs. You only miss the delay configuration, so there's a good chance that one/some of those switches and/or VCAs have packed in. If so, that would likely be a very localized fault that a tech doing component level service could fix fairly easily.
I'm sure Arturia will help you to get a service arranged. Crossing my fingers for your delay FX soon being back in action!