Export my photos from photos app on Mac Ventura 13.0.1 with date taken included in metadata.

Dates can be a problem. When you use "File>Export> Unmodified Original," the date will be the taken date, but the file will not have any changes, edits, titles, keywords, etc, that you added

When you "File>Export>Export xx Photos," the date will be the export date, but the pictures will have all the edits and additions. That makes some sense, since edits and additions have changed the file, but it's not what most of us really want.

Probably the best solution is to use the 3rd party app, PowerPhotos, which costs maybe thirty bucks, but gives you a dialog window like this when exporting:

It lets you choose the date you want. Luckily PowerPhotos has lots of very useful features that help with Photos. Lots of people here use PowerPhotos, and I keep it open all the time when I use Photos.

There are other 3rd party apps that can change the metadata on the pictures that have already been exported from Photos, copying the capture date to the file date. I use GraphicConverter for that. But there aren't any other options available within Photos.

When you export a photo as a jpeg the file creation date will be the export date but the EXIF image creation date will not change.

To test it export a photo as a jpeg and open it with Preview. Open the Inspector and select the EXIF pane. There sill be the image creation date which will be the same as the original.

This photo was taken Oct. 12, 2021 and exported today:

You could also use a subfolder format when exporting from Photos.

When you use these settings when exporting the photos, the photos taken at the same place at the same time will be grouped by folders named like the place and the date. Tat makes it easy to sort them and find them again.

I have changed the date format in the Terminal to year-month-day, so the folders for the same location will appear in a chronological order.

For example: