@tizzo 

It may be the drive is pulling to much power. Consider using a power hub.  
 

Does a usb stick work okay?

  @ArcherC8 Hi, thanks for the response.  The drive enclosure is externally powered, so I don't think that's it.  I've got some other drives to test with, and will try them out over the next few days as time permits.

  @tizzo 

What format is the drive? The router does not support all formats.

  @ArcherC8 You mean the file system?  It's NTFS.

  @ArcherC8 Do you happen to know what filesystems are supported?  NTFS is not my first choice (but it's better than FAT32), but I could not find a definitive list of which are supported.

  @tizzo 

If you click on the model in your orginal post (or google the model number), the TP Link router info will show up, then click on support, then you should see a link for the user quide. Note that some of the models, you need to go to the first version of the router to see the user guide.

It shows the following in section 9:

If you use a USB hard drive, make sure its file system is FAT32, exFat, NTFS or HFS+.

  @ArcherC8 Ah, thanks for that info.  All I had found previously was a generic page saying that some supported FAT32 and NTFS, and others supported those plus the other two.

Anyway I think I got to the bottom of my issue.  Your theory about the power consumption was probably the closest. . .

When I disconnected the drive and plugged it into my wife's PC to run chkdsk on it, I noticed the connector was a little loose in the USB port, so I switched to a different cable.  After running chkdsk I put the original, shorter cable back on and reconnected to the router, and nothing changed.  So I swapped back to the the other cable, tried again, and it came right up and has continued to be available, for about 6 hours and counting when I left the house this morning.

The original cable was USB 3.0 (as are both the router port and the drive enclosure), and the only others I had were all USB 2.0, so I don't know if the cable is actually bad, or if myabe the drive can't handle USB 3.0 speed and the enclosure isn't handling that well.  Seems unlikely, but it would explain why I didn't have this problem with the same cable and the old router, which I think was strictly USB 2.0.

Anyway bottom line is I think I'm good, thanks for the support.  The rest of the details are included only in case they are helpful to someone else in the future.