Did The Cosby Show become dated the moment Roseanne went on the air?
To put things into perspective, The Cosby Show was about a family where the dad was always right, the mom and dad always had wise and pithy advice for their kids (or a funny joke), where the kids recognized the wisdom of their parents and where no one ever really made a meaningful mistake. In essence, The Cosby Show was the epitome of happy family sitcom to that point.
Roseanne when it first came on in 1988 however, blew it up (as did The Simpsons and you can argue, Married...with Children). The Conners in contrast to the Huxtables, were a family that featured people who didn't like each other at times, were ignorant and flawed and made bad decisions. Also as partially a consequence of those decisions, the Conners didn't have a great life - and yet they were also a family.
This is probably why Roseanne seems to hold up better and is funnier than The Cosby Show because it addressed much harsher realities.