Missing photos from August 9 on iPhone 16 Pro Max

It could be that you aren't looking at the right place. For instance, it might be that on August 9 that you were in a different time zone, and so the pictures have a different time than you expect.

You can see this:

         If you're missing photos or videos in the Photos app - Apple Support

Basically the above says to look in Recently Deleted and in Hidden. Be sure to check your Library thoroughly. You can use the search field with the date or location. And remember, you can sort the Library by date captured or by date added. Also, your Library view now has filters, and you may not have your filter set to "All Items." The Filter is located in the same menu as Sort:

If you don't use iCloud Photos, but you have Settings>Apple ID>iCloud Backup turned on, then you may be able to recover data from the last 30 days or so. You should look here:

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support 

“Restore” is what you do when you buy a new phone, so this safety, if you have it set, requires you to erase everything on your phone and replace it with the backup.

If you don't keep your own backups to an external drive, then there really isn't any place else to look.