Is “curate the market” common usage of “curate”?

My reading of this is that the publisher, through the books it has previously chosen to publish, has shaped the market. If a publisher is acting as 'curator' (by choosing which manuscripts it deems worth publishing), it is indirectly curating the book market.

To answer the headline question directly: no, it's not common usage.

That is a rather odd use of the word "curate" in the verb form, as their particular usage may change what they intended the sentence to mean. I doubt that they meant to imply that the publisher has great power over the market and actively controls it, rather than merely participates and contributes to it.

Also, the definitions that you found for "curate" are probably a result of poor website design on behalf of those dictionary makers. The first definition should be for the verb form rather than for the noun; and in fact Oxford Languages says that the noun version of "curate" is archaic, but they still list it first. The better definition is for the verb form: "select, organize, and look after the items in (a collection or exhibition)." Those dictionary webpages now do list that additional form of the word and the definitions on the entry for the word, though it is lower down.

Regarding if you should use "curate" in the way they did, I would suggest against it. Curate is most commonly used when talking about a manager, supervisor, overseer or maintainer of a collection or some form of historic or artistic entity - often a museum. Though you could say "you must curate the offer letter that we will send to the applicant we selected" - to imply that they should take great care when writing the letter. A poet is the curator of the words they put in their poem. This stretches the definition, but it is respectful to the formal definition, and is used to emphasize the care that the person puts into the work.