The short answer is no.
The cable port (coaxial) on a TV doesn't have that function. The cables port's primary purpose is to interpret a video/audio signal from a source, in this case, the cable provider. Think of it like an HDMI port, but lesser video quality. The ethernet port on the other hand is meant to create an ethernet connection and in this case to create a connection to the cable provided network, through a modem. Coaxial cable can provide the connection via the wire, but there needs to be something to interpret that signal.
You will need a cable modem or cable modem/router combo to "interpret" the signal and router to provide wi-fi or be able to hardwire from the router to the network port (ethernet) of the TV. I personally recommend the Roku devices with any name brand TV instead of the TVs as there seems to be less problems with those and if something happens to the Roku box, then that cost less to replace than if the TV becomes obsolete or has a problem. I also recommend a separate modem router than a combo modem/router device as those work better and provide a stronger more reliable signal.
You could hardwire the TVs to the network by getting a router that has a mesh network with satellites (like the Orbi or others) and connect the TVs via network cable (Cat 5 or 6) from the router and satellites to the network port of the TV. Wireless is easier, however, how reliable the signal depends on a lot of factors including distance from router, how big your house is, what the house is made of, how crowded the wireless signal is, the quality of router you have and other things.
For example, I have Roku Ultras instead of cable boxes (using Spectrum cable and internet). I still have cable service but save by not paying for boxes. If you get rid of your cable service and just use the internet, this works basically the same. I use the Spectrum provided cable modem connected to an Orbi router and satellites, which provides the signal to the Roku whether that is the Roku box or TV. Some I have hardwired direct to the router and others I do not and are connected wirelessly. All of the Roku devices work equally well as far as signal and reliability. Now you don't need to spend as much as an Orbi costs as there are cheaper mesh networks out there, but of course that depends on your budget and features you want.
The other option of course to connect "Cable" is using the Comcast provided cable boxes via the old fashion way of cable TV. I think your wife is confusing the terms a bit. You can get "cable" internet but not in the sense of cable TV with cable boxes. That is the purpose of a modem in that instance.
If you want all of the TVs or Roku devices hardwired into the network, then you need a way in each room to access that network via ethernet cable either by having a router/extender/satellite in each room to hardwire to, ethernet cable run through the walls in each room from the router or server. Or you can connect each of those wirelessly from the router. Just make sure you use a secure network with a good password. Hardwiring those rooms for ethernet is going to be more expensive to do, regardless of whether you use a mesh network or try to hardwire through walls.
