Dear William Miles This is a new player in the field of scientific publishing. I think it is not a black or white case when one look at this publisher. I see good and somewhat disturbing things.
Somewhat disturbing:
-I cannot help it to think that their logo is ‘inspired’ by another well-established publisher (see enclosed figures)
-Looking at a journal which is somewhat more in my own field of expertise then I see a review which is somewhat sloppy and more importantly it miss all the references in the paper(text) https://www.anserpress.org/journal/bab/2/1/7
-Looking at the journal “Journal of Economic Analysis” I see that this year they publish papers that where already accepted quite a while ago (some took almost year), see for example https://www.anserpress.org/journal/jea/3/3/63
-The same is true for the journal you refer to see for example https://www.anserpress.org/journal/eal/3/2/52
These are all not extremely serious issues and probably due to the fact that they run the operation basically without income at the moment.
The comforting things are:
-As said, they publish this year without charging an APC https://www.anserpress.org/page/publishing-fees This is not the behavior of a typical predatory publisher
-Looking at for example “Economic Analysis Letters” I see in their indexing info https://www.anserpress.org/journal/eal/page/abstracting-and-indexing genuine metrics (J-Gate, Dimensions, CrossRef) and no mentioning of so-called misleading metrics (https://beallslist.net/misleading-metrics/ ) these are often used by predatory journals/publishers
-Their editorial team look legit (https://www.anserpress.org/journal/eal/editors )
I estimate them not as predatory, and I think there are genuine people behind this legit looking initiative. So, I think it is reasonably safe to seriously consider agreeing to their request to review a paper for them.