I thought I had a clever solution. Seemingly the Roku CBS app is at fault.
So, as I own an Amazon Fire smart TV - I used the Amazon CBS app, connected to Verizon, and was streaming live. And, silly me, I thought it would persist past midnight.
But, nope, the Amazon app behaves the same way as Roku. After midnight it requires me to re-connect through Verizon.
This re-connection process is irritating. It requires another device to go to the CBS page, and then log on through Verizon after entering the code produced by the Roku/Amazon app.
It's not a big deal to do this occasionally, but to need to do it every day is more trouble than it's worth to me given the cable box requires none of this.
So, I guess I'll keep the cable box. And I now think it's likely the case that CBS enforces this policy.
As an aside, note that ABC and NBC don't behave this way. And while all three of them (CBS, NBC, and ABC) only live stream our local network, the FOX app actually allows you to chose from a wide variety of local fox channels across the US.