Maybe your services relied on the "supercookies" thus being immune to 3rd party cookies disabled :) ?
An example that I can imagine is a big onlite shopping company that has several domains, and they want a shopping cart that that works across all their domains.
A big online shopping company that moves you between several domains as part of normal use? tbh I've never seen a single site do this, so I don't feel bad saying "no, I don't think that's a legitimate reason to allow 3rd party cookies".
Besides, since they control all the domains, they can do redirects to associate your session across them easily enough. It's basically oauth at that point, which works just fine without 3rd party cookies.
An example that I can imagine is a big onlite shopping company that has several domains, and they want a shopping cart that that works across all their domains.