Why do we need Gibbs sampling (and MCMC)?
It's not "essentially $O(1)$" to draw objects from a set with non-uniform probability: your bucketing scheme takes more than constant time.

Further, sampling from a Markov chain allows you to sample without having to construct the probability distribution explicitly or even the state space, explicitly. For example, how do you propose to randomly sample matchings in a graph by your method, even uniformly? You'd need to first construct the set of all matchings (of which there can be exponentially many), then select one. And note that constructing the set of matchings implies being able to count them, which is #P-hard, so very unlikely to be doable efficiently. With a Gibbs sampler, you just run a Markov chain for polynomially many steps and you have yourself a matching chosen uniformly at random.