The main argument against direct democracy is the danger of majoritarian oppression: e.g. imagine calling a vote to reinstate slavery and succeeding because 51% of the population supports slavery. Having a set of laws like the Bill of Rights that is agreed upon by consensus and enforced by courts is a safeguard against that.
The main argument against, as I see it, is the lack of accountability, the prime example being California demanding more spending and lower taxes at the same time. Of course, indirect democracy has done very little to prevent that either.
When Switzerland doesn't pass laws similar to California's, I think it's because they have a much more homogenous society where there is simply a large degree of agreement on how things should be done - and so, the remaining bits lend themselves well to be decided by direct democracy.
That's also my response to the GP: no, no amount of interwebs and technology can turn a large and diverse population that disagrees fundamentally on many important issues into a small and homogenous one that agrees on most things.
The Swiss aren't homogeneous, though. They have four official languages and 20% of the country are resident foreigners. The various cantons don't often agree with each other and only 30-40% of the country votes.
How likely is it that a majority of a modern society would agree to something crazy like this, vs. just the ruling party of an indirect democracy? Unfortunately we don't have much statistical evidence, since there's basically only Switzerland having such a system nationwide, but at least we know that in the history of this republic since 1848 there hasn't been such an incident, while the Weimarer Republic, culturally close to a majority of Swiss and under an indirect democratic rule, managed to elect a ruling party that went haywire.
