"Does anyone know how LR can export that info (I have "All Metadata" checked in the Export options)"
LR does indeed export the capture date to the EXIF:DateTimeOriginal field. You can verify that by reimporting the exported photo into the LR catalog, by viewing the file's metadata with Exiftool or other metadata tools, or viewing the Date Taken column in Windows File Explorer.
If those methods aren't showing the capture
date in the exported file, we'll need more details.
I think I may have come across a bug in LR, but so far I haven't been able to replicate it.
I started out importing 1268 raw images taken with 2 Nikon D810s and a Sony a6000. There was no chance of filename conflicts because of file name extensions (ARW, NEF) and sRGB vs Adobe RGB on the Nikons. Then over the course of 3 weeks, I edited the files very much in a random order, starting with easy edits and progressing to harder ones. I started with basic things like Auto WB and exposure with some sharpening and noise reduction where it was needed and cropping the images as I wanted. The images I was going to export as 8x10 I put into the Quick Collection, others where 8x10 wasn't feasible were rated 5 stars, and still others that I wasn't going to edit yet but still might keep given a red label. I then filtered on the QC, selected them all, and exported to a specific folder (asking what to do with existing files), using the same filename with an uppercase extension and 3000 pixels at 300 PPI along the long edge. I then marked them as rejected ("disabled" thumbnail as a quick way to see it had been processed) and cleared the QC. I did the same with the other categories, choosing a different folder and long edge pixel count of 5400 for the "unfeasible" 8x10s, or unchecked "Resize to Fit" and yet another folder for the unedited ones. All other export options remained the same. After each export, the filtered photos were rejected and the attribute removed. In each of those exports, it was highly likely that images from all three cameras could appear in the filtered set. I ended up with 960 images in the final catalog where the other 308 were removed from the catalog and deleted off the disk because there was no chance they were going to be used.
So today, I selected a few of the rejected 8x10s, exported them, and lo and behold, the Date Taken appeared in the Properties box for each file in Windows File Explorer! I thought perhaps the checked items in the Metadata section might be the issue or perhaps a different destination folder I used for my testing but nothing manifested the issue I initially found. Believe me, I would have never submitted this issue if the dozens of files properties I looked at all didn't have a missing Date Taken.
I'm going to try the export again, except this time I'll be choosing the files for an entire category so there's pretty much a guarantee that each one will have images from multiple cameras, different export location, and different sizing options. The hard part will be ensuring I select ALL the files in each category from the 960 before doing an export.
If I come across this issue again during those exports, I'll let you know and if it still exists for my "unique" circumstances, perhaps you can let me know some things to try to help isolate the problem. If all goes well, I'll let you know that, too.