In August, we had trouble with the Tablo “forgetting” our paused/stopped marker and was starting back at the beginning (Yes, I know there is a fix for this now, and I’m glad, thanks to the Tablo guys for the fix). Since we had never tried a Roku, we got one a couple of weeks ago, and installed it on the same tv as the Amazon Firestick. The Roku will access the Tablo recordings just fine, but it stops to spin every few seconds, and spins for 10-15 seconds. It’s maddening and nearly impossible to watch a recorded 30 or 60 minute tv show (and a football game is not even remotely possible). I’ve tried all combinations of rebooting, from routers, to the Roku, to the Tablo itself, and no matter what recording I choose, it starts spinning and buffering on watching it. I’m sure this is an easy fix, something I’ve setup or done incorrectly, but I cannot for the life of me figure out what to try - so any and all help would be greatly appreciated 
What Roku did you purchase? The cheaper ones have less expensive processors. Is the Roku wireless or connected VIA Ethernet?
It’s the Roku Streaming Stick+ ($60 at Sam’s Club), and it’s connected wirelessly. All of the other services we use work great, but every recording from our Tablo DVR (that work fine via the Amazon Firestick), just spins every 10-15 seconds. It’s virtually impossible to watch anything from our Tablo 
I had a similar problem with my Roku 4200.
Mine was connected wirelessly on the 2.4GHz band for several years without any issues. Then one day it started exhibiting the same problems that you have described.
My solution was to change the Roku to the 5GHz wireless band. It works fine now.
I concluded that the 2.4GHz band in my neighborhood got over crowded to the point where the Roku was competing for bandwidth.
Hope this helps with your problem.
DrJack - thanks for the feedback! We have tried that and it doesn’t seem to make a difference. It still just spins every few seconds and makes playback of our recorded shows almost impossible to watch