Risks of shifting to post-publication review
In my mind, there are three semi-related problems with post-publication peer review:


It's difficult for the lay press to distinguish between pre- and post-publication reviewed papers, and for many "breaking" articles, I'd be concerned about reports of spurious results becoming more common, not less. For example, if I publish a paper on a risk factor for X disease, how long does the press have to wait to report it? Do they have to let peer reviews build up? Pre-pub at least ensures that someone has looked it over.
Article readership is nowhere near uniform, and there's no reason to believe reviews would be different. At the moment, every single paper published by a journal is subjected to some form of review, whether anyone else cares about that paper or not. Relying on post-publication, it's possible some papers will get few/no reviews. What's a non-expert to make of those papers? Similarly, what's a non-expert to make of a post-pub vaccine study that's been flooded with anti-vaxx rhetoric dressed up in technical jargon?
Post-pub peer review is inherently non-anonymous. I'm pretty sure Big Deal Figure in the Field is not going to be judged as harshly as New Grad Student even for the same work - and this will also likely effect those with clearly female or minority "sounding" names. Prominent scientists may be more likely to get a pass under the assumption that they're  right. In my mind, breaking review anonymity is a step in the wrong direction.