Does anyone know how easily you can do "copyright clearance" for these supposedly public domain images?

For example, the page for the first image I clicked on said:

    Date
    1833

    Underlying Rights
    Public Domain Worldwide

    Digital Rights
    No Additional Rights

    * Source states “no known restrictions”
    * We offer this info as guidance only
with a link to: https://pdimagearchive.org/reusing-images/

If, for example, you design the cover of a self-published book around such an image, is Amazon KDP going to reject it, because they don't accept that screenshot as sufficient proof of rights?

Search for public records about the image. If you can confirm the author is dead for at least 70 years (in case the author originally held the copyright) and the work is at least 120 years old or was published more than 95 years ago (in case this was work for hire) you are good

For younger works other conditions might make a work public domain, like the work being created by a US federal government employee as part of their duties

Just contact the original author. You may need to pay extra for the Medium.

I know this is a joke, but just as a note, in some european countries, the person who digitized the artwork may have a copyright. It varries a bit by country.

Exactly. The site mentions that and some other problems, and just disclaimers their liability. One problem I didn't see mentioned directly is that some venues have additional bureaucratic rules.

For example, I saw a writer-artist complain that they couldn't show the big ebook store proof that they'd licensed the art for their cover, since they painted it themself, and there was no documentation licensing it to them.

So one could use practical advice from someone who's figured out the rules, and has the battle scars, like: "With public domain art, just make sure you do X and avoid Y; otherwise, there's a 50% chance that a random copyright check the first week will result in your book temporarily being shadowbanned, and you won't get the crucial early sales while the algorithm is trialing exposure of your book in searches and categories, and then the algorithm will pretty much never show your book to anyone ever again."

Fwiw, i think wikimedia commons has some really good help pages on what is and is not public domain:

https://commons.wikimedia.org/wiki/Help:Public_domain

https://commons.wikimedia.org/wiki/Commons:Freedom_of_panora...

https://commons.wikimedia.org/wiki/Commons:Reuse_of_PD-Art_p...

https://commons.wikimedia.org/wiki/Commons:De_minimis

Of course sometimes its just a risk trade off, and depends more on how the publisher feels then what the actual rules are.

What I'm wondering is -- although an item would be found to be legally considered public domain in all applicable jurisdictions when examined by law clerk -- what are the practical rules for passing copyright clearance checks/challenges by Amazon, Kobo, etc. for public domain art.

You don't want a book's carefully choreographed launch sabotaged by various companies' Kafkaesque bureaucratic processes, while you're trying to finesse the algorithm and the publicity you've scheduled.

(I should've said upfront that I know a bit about public domain and copyright law as a layperson. And I also know that I can't necessarily point a CSR to law surveys if something gets flagged. I need to know the "do X and avoid Y" that is proven to actually work smoothly with all these companies' ideas of "copyright clearance", so that I hopefully never have to talk to a CSR.)