The link below is obviously a "shill post" from the makers of "phone recovery software" trying to sell it to you, but they have a page that talks about how to recover from the "Black Screen of Death".
[Solved] How to Fix A Samsung Tablet with Black Screen
"Usually on a phone that packs a removable battery, you can take out the battery and put it back in to solve the problem. But since tablet doesn’t have a removable one, this does not work for you.
In this case, just make sure that your tablet is completely turned off and then boot the device into recovery mode by holding Home, Power and Volume Up buttons at the same time for a few seconds. Then you can solve the issue manually."
Similar post on the Samsung Community Forums with a slight difference.
Galaxy Tab S3 will not switch on!
They say the same thing but say to use just Power and Volume Down for 7 seconds.
Even better, I found a confirming post on ifixit site. I trust ifixit a little more myself...
Galaxy tab 3, screen has power but is black.
This solution says to use Power and both Vol + and - at the same time.
I don't have this tablet but am thinking about getting it (less so now that I found this thread (Black Screen of Death? Yikes!)). Posting this info here on the hope that it helps someone else with this issue. If you try this solution, *PLEASE*
post a reply and indicate whether it worked or not so others know if this solution is worth anything or not.