I have 5 TV’s in the house. Each TV has run of two Cat6A cables connected to the central switch in the utility room.
Do I need one tablo for each TV (5 tablos)?
No, you need one Tablo. Then you need some sort of streamer box (Roku, AppleTV, FireTV) or a smart TV at each location. See this page for supported devices:
and start with the basics
https://www.tablotv.com/tablo-dvrs-how-they-work/
There is great misunderstandings, needing a tablo per TV.
As long as you can upload a Tablo app to your device you can watch TV. This includes phones and Tablets. You can watch your local TV on your phone from France as long as you have a good connection. You’re only limited by the number of tuners in your Tablo.
In my opinion, the best way to view and record over the air TV.
and a subscription for Tablo Connect 
With that many TVS you should buy a 4 tuner tablo.
Number of tuners matters mostly as to how many channels you get… and actually record. If using the tablo, how many users watching Live TV simultaneously.
A single 2 tuner - Dual Lite, can accommodate (up to) SIX streams