Switching to server side storage would've been a too costly architectural change. What if the server side was stateless before?
And even if, web applications would still have needed to support the old model for interop reasons. It's not feasible for everyone to maintain code for both approaches when they are so different.
It'd also break many use cases, such as using a (signed) cookie as cache for some really frequently used data. Or more importantly authentication, where an auth service returns a token (in some cases a token that can be checked by a different service without contacting the auth service again).
So yeah, that was a non starter.