But does not having comical relief necessarily make them boring? Team Rocket may have been entertaining during OS, but the act got stale, especially by the end of DP. If that would have continued during BW, I think most fans would not have found their relief to be comical anymore. They were due for a change by the time BW aired, but whether or not their transformation into seriousness was the best direction, any change would be something that would have impressed half the fans and annoyed the other half.
To be honest, I wasn't exactly happy with how TR was handled either pre-BW or during BW. At least with the OS, in addition to having some comedic moments, they actually were played as genuine threats that the main protagonists actually needed to stop most of the time. In fact, I'd argue that keeping them in AG was a huge mistake, bigger than even their removing Misty, since from AG onwards up to BW, their actions were just pointless, very rarely did they even pose a genuine threat to the heroes, most of their actions could easily have been done by the villainous team of the region anyhow (heck, have Team Magma pursue Pikachu because he was a suitable host for the Red Orb, especially considering the events of the Groudon and Kyogre two-parter), and they came across as annoying and stupid, not to mention their disguises were so bad by that point that it only seemed to serve to make Ash and co. seem stupid, especially Ash and Brock who should know better. Even when we got to BW, where they admittedly did get serious, they did it in the absolute wrong manner, often times their serious plans came across as poor planning. Take the Meowth mole arc, for example. There were several episodes where it was treated as though Meowth actually did end up sacked by Team Rocket, and even Jessie and James confirmed it in a private conversation between themselves when there wasn't any chance of them being listened in by the main protagonists, yet then it turns out Meowth was actually a mole all along. Even Dogasu commented that arc was just poorly thought out. There were plenty of ways to hint at Meowth actually being a mole for the viewers without making the rest of the cast seem like idiots. Take the Dawn of X arc from the Mega Man Archie Comics serial. It's made obvious from Dr. Wily's private thoughts that he's intended to screw Mega Man and his allies over, yet he nonetheless is also depicted in a way that has everyone in-universe not suspect him of causing trouble for understandable reasons. If TR is to have comedic elements, fine, but at least make sure they actually are credible threats so they actually have a point towards being villains instead of just plucky comic relief characters. General Blue from Dragonball was extremely comical (heck, in the FUNimation dub, he even sounded a bit like James), yet even he proved to be a very dangerous threat to the main casts despite his comical demeanor (heck, Goku couldn't even beat him in a fair match without assistance thanks to Blue's psychic abilities).
As for the main topic, I have a hard time seeing how the anime would be without Team Rocket. A lot of plots depend on the actions of Team Rocket, but I guess it wouldn't be hard to replace them with wild Pokémon or other mischievous Trainers in that sense. But the lack of their development and their own stories and history would surely be missed and would take some depth out of the show as a whole.
It probably would have worked with AG onward, not sure about the OS, though.
