The author suggests that "The fear that multiple submissions would overwhelm the peer-review system lacks empirical evidence". Maybe it won't "overwhelm" it, but it will certainly add to the reviewing workload. Simply stated, if authors submit to N journals and each asks for 2 reviewers, that's 2N units of work (assuming they can get the reviewers), compared to 2 units of work.

But it may be worse than that, actually. I will not be much inclined to bother reviewing, if I know that the authors might pull their manuscript if another journal gives a green light quicker than the journal for which I have been asked to review.

The solution to a slow reviewing process is not to ask reviewers to do more of this unrewarded work.