Photos taken with the Moment app only show Date Time
That's the problem --- the Moment app isn't playing by the rules.
The EXIF DateTime field is confusingly named. Under the EXIF standard, it is the date and time that the photo file was last changed by standards-compliant photo software. The EXIF DateTimeOriginal is the "capture time", when the shutter was pressed.
Based on what you're seeing, the Moment app isn't adding EXIF:DateTimeOriginal to the photo. If true, that's a huge mistake. Firmware and software developers are often sloppy about following the standards, but omitting DateTimeOriginal is much worse than normal.
If you want to double-check and gather evidence for a bug report for the Moment team, take one of the original Moment app photos (not touched by the iPhone Photos app, LR Mobile, or LR) and upload it to Dropbox. We can put it under the microscope to see what might be going on.
I just took a photo with the iPhone using the Moment app and placed it in a Dropbox folder. The link to the folder is Dropbox - Moment Photo. Thanks for looking into this. I also sent me query to Moment, and I will add your concern about Original Time.
The link to the folder is Dropbox - Moment Photo
I just confirmed with Exiftool that the sample file doesn't contain EXIF:DateTimeOriginal, though it does contain EXIF:DateTime.
You might send the following to Moment, from the EXIF 2.3 spec:
Thank-you very much. I appreciate your confirming that the problem is with Moment and I will forward the information you sent me.
I hope that the people at Moment will be able to correct this.
Thanks again, Linda