It would be good if you described your signal chain better.
I'm guessing based on your text your signal chain looks like: roku->HDMI to Composite audio/video adapter box-> ATSC DTV tuner box-> TV set.
That setup, if it's what you are running, should NOT work at all. The ATSC DTV box's RCA jacks are OUTPUT ONLY. You should never try to back feed signal into them from another signal source device (ie don't use them as an input because they are NOT an input).
Replace the DTV box in that chain with an RF modulator to make the Roku work...Or if you have multiple HD and SD TVs in your house that you want to share the Roku, and don't mind spending $200-300 to make that work, you could get a ATSC DTV modulator (pick one with better than 40dBmV output), plug a UHF antenna and a Roku into it then tune it's OTA signal on your DTV tuner for the Zenith and on your HD TVs...I ended up doing a more complicated version of this with multiple HDMI source devices through a switcher box for various reasons that I explain in these 2 videos.
https://m.youtube.com/watch?v=hKjt3x4WtWU
https://m.youtube.com/watch?v=j1iN4ULZcX4
If you go with the more affordable option of just swap the DTV converter box for an RF modulator you will probably be annoyed that you have to unhook the TV from the modulator and switch it to the DTV box and vice versa when you want to switch source devices....My recommendation is to get a composite audio video switch box feed the RCA outputs of your DTV box into one input and the RCA outputs of your HDMI to composite adapter into another input then feed the output of the switch box to your RF modulators RCA input and use the switch box.
One thing that sucks about folks my generation and younger is that unless you grew up with older component stereo systems (like me) or your family was into home theater/game consoles and let you wire stuff for that the concepts of one way analog signal flow from jacks and inputs, outputs, splitters and switchers are all alien to you and need to be understood to prevent miswiring a signal chain and or multiple component video source system.