Cable and Sat remotes usually have an "Input" button on them. That would eliminate the need to use the TV remote for that function. The Cable and Sat remotes will control most current models of TV's if you program them properly with the best make and model info or code number for your TV model. Some units or remotes are easier to program than others. In all cases you can:
1. Go to the Cable and Sat company's website for instructions and codes
2. Go to the Cable and Sat company's Support - Chat or email
3. Call the Cable and Sat company's support phone number and speak to tech support
4. Have a tech come to the home/office to program the remote for you. There may be a charge for this
Additionally, in the Cable or Satellite interface - go to - menu > settings > HDMI CEC, make sure HDMI CEC is turned on. HDMI CEC enables the SAT or CBL button on their remote to power on your TV and change inputs.
If you would like to improve your TV experience a little more, read on:
Cable and Satellite receivers get periodic Firmware updates through their connection medium (cable, sat signal, phone line, etc.), but TV's don't, unless they are Smart TV's which are connected to a network and there is an auto-update function. Updating the Firmware on your TV can help a variety of issues, which may include connections. This update might help the Cable and Sat remotes more accurately control the TV. If you had previously used a generic code for your brand of TV with your Cable or Sat remote, you can now try reprogramming your remote with more recent updated codes.
Contact your TV's Manufacturer for more information on updating Firmware using one the 4 steps listed above. For more complete information, I would go to the Manufacturer's website and download a PDF version of the user manual, also the Tech or Admin manual, if available.
Search/Google Tom's Hardware and other websites for codes too, as I've occasionally found better code numbers than what the Cable and Sat websites offer. Tom's Hardware members can help with Firmware questions too.