> July 1, 2021 issue

I guess that they published the July issue some days early. I guess that if the month issue is already finished, it makes sense to publish it.

Here you can see the mail-date cover-date relationship for the physical publication: https://www.nybooks.com/publication-schedule/

The magazine with cover date "July 1, 2021" is mailed on "June 8, 2021"

It's common for every publisher to publish things early. When I worked at a publication, the July edition would hit stands in mid-June. Same thing happens across the industry.

Magazine dates are typically the date the issue is removed from shelves, not the date it first appears. So they likely synchronize with the actual date of print availability.

Magazine issue dates are usually for the month they are on sale, so realistically a month ahead of when you'd buy it if you're buying it as it comes out.