[error] on install (the 'core' key in *.info should not be set after Drupal core ^8.7.7)

Obviously, the core key is for installations before D8.7.7. When you don't want your module to be installed/updated on those environments you should remove the key. When you allow it on those environments you need the core key. Don't know what causes this error, if it is core, you might try ^8 instead of the more logical ^8.7.7.

It appears they consider this error to be correct and are not prepared to change that. So, either change the version constraint or change the code, preferably by patch so you can repeat it.

Nah this is a major bug, you cannot install a new site.

As the comment says, the dependencies key below can/should be removed too.

Also, a drupal/core requirement should be added to composer.json as drupal.org doesn't do that yet automatically.

This is also removes the no longer required dependency and adds it to composer.json, which is currently required to fetch it with composer as d.o doesn't respect the new key yet when generating that.

I vaguely remember that Drupal core 8.7.7 was complaining when I had the following lines in the .info file:

core_version_requirement: ^8.7 || ^9
core: 8.x

I'm pretty sure it was complaining that I couldn't specify "^8.7" because 8.7.0 doesn't understand the core_version_requirement. Which is why I changed it to "^8.7.7" even though the module is compatible with 8.7.0 and later.

Anyway, the patch looks good to me.