If you don't have burn in, you bias your estimate. Sure, in the long run the bias becomes smaller and smaller, but, as the saying goes, in the long run we are all dead.
MCMC is generally slow, any way to speed it up a bit is good. Throwing away the burn-in part of the chain is just such a way to speed it up.
My intuition is you want to be somewhere in the middle of the tradeoff. You want to tolerate some bias -- use burn-in to get partway but only partway to the stationary distribution. Here's a paper that advances this way of thinking: http://web.stanford.edu/~ngoodman/papers/LiederGriffithsGood...
However generating an approximate sample is the exact problem that a MCMC process is trying to solve.
If you have a chain, and throw out the beginning fraction, is that any more burnin than starting with X_b, which you probably had to find using some period of... burnin?
This is really all about time series heterogeneity which is a topic unto itself. Eventually the chain reaches some stationary distribution but in the meantime there will be some shift in the distribution potentially. Seems like you could model the chain to do some test of where the distribution shifts, if at all.
I imagine this is in the literature. Giving it a brief look I can't find anything but I haven't looked hard enough.
In order to get a reasonable answer from MCMC, you need to run for many correlation times. Burn-in should only last a few correlation times, so you don't lose that large a fraction of your chain by throwing it away. And the part of the chain you're using is biased, so you're probably not gaining anything by keeping it.