I love these threads. They never fail to amuse.
IMO there are two sides to the argument but for some reason neither party concedes this. One is right, not both.
Well IMO both are correct.
Cycling alone will not provide the best possible core from which the power will ultimately derive from. Core specific training is not done on the bike. Core training is widely known to aid power transfer and cycling efficiency, not to mention helping the ability to endure serious time in the saddle.
One could also argue, though not proven that i am aware of, that a cyclist can only train the muscles directly related to cycling. The result being that the cyclist is prone to muscle imbalance and/or future injuries that might otherwise be avoidable with a more rounded training program. Some cyclists add running etc to their training to prevent bones becoming brittle and have a more general fitness rather than cycling specific. Specific stretching also can be applied to ones training. Little of the above is "weight training" but it is not just "riding a bike" either.
End of the day though one could manage to become a very good cyclist just through cycling alone, with not one other bit of training at all. I don't think that person would be reaching their potential as an injury free cyclist or even utilising their power most efficiently, but that's just my opinion. It just happens to be fortunate for me that it also happens to be the similar view of the "strength and conditioning" GB cycling coach. The same guy who has all the cyclists in team GB partake in core strength training and adding dreaded "weights" to their schedule. Not just the track guys either.
FWIW, i ONLY do core training and do not subscribe to the view that weight training your legs will make you a better/faster cyclist. It MAY, i speculate, help to better balance muscle groups, if done correctly, but i do not personally find this a worthwhile or desirable end product.