Why and how commonly are scientists concerned about open peer review?

This answer discusses a survey that indicates blind reviews are much preferred to open ones. The main reason, from that answer, seems to be

• Many people don't want to review non-anonymously. I know many people on both sides of the fence on this issue. Once concern is that authors are often sensitive to criticism and may harbor ill-feelings against a person who gives an unfavorable review (especially if the reviewers have not properly understood the paper). Consequently, open reviewers may be biased (more so than blind reviewers) to colleagues they respect or want a good relationship with.

Fomite mentions a couple of other issues, but I see this as the main one (and it seems the survey does too). Some other potential issues are

• Certain "rivals" or "critics" can try to ruin your career by giving unfairly harsh reviews.

• Some well-respect person could make a comment based on a cursory read of the paper that ends up heavily influencing the reviews, and maybe no one reads the paper carefully enough.

I don't know that there are any surveys looking at this, but some concerns I have, or have heard from peers, involve the following:

• Open means open, which results in a way more expansive definition of "peer" than in other journals. This can range from the merely problematic (people outside your field ignoring field-specific convention) to crippling (mass commenting on controversial topic by trolls).
• It's the antithesis of double blind peer review, which is also something that's being pushed fairly heavily. If you favor that approach, post-publication peer review is something of a problem.
• Post-publication peer review is not fairly distributed. J. Random Student isn't necessarily going to get the same kind of comments as Professor Big Deal. It's possible they'll be different in volume, hostility, etc. Standard peer review, for better or worse, does ensure a set number of reviewers.