You need a coax input to get the RF signal to the tuner. Tuners do not accept signal via component, composite etc. They get it from an RF connector sometimes called a coax input. A tuner would be worthless without one. No offense is intended but I am missing in your logic as well. Why would a set have a tuner if it had no way to get a signal to it? Perhaps you think that the component or composite cables are sending RF signals but it does not work that way.
If there really is no coax input it is a monitor. No problem with that. if you plan to get TV from other sources it is fine but know what you are buying. If you intend to use OTA signals the monitor is not right for you unless you have an external tuner of some kind to feed those other inputs.
But a set without a tuner is called a monitor. And there would be no reason to build a Tv of any kind with a tuner and no coax input. Neither of those things have any use without the other.
I have a first generation HDTV with no internal digital tuner. It has an old analog tuner, but no digital tuner. For digital, we used a digital box from our provider. If I wanted to use digital OTA signals, I could have bought an expensive HDTV tuner. My two newer HDTVs have an internal HDTV tuner. In either case, they still called my TV a TV.
A solution here if this thing does not have a coaxial input would be for the OP to use either a VCR or DVD-R recorder or combination of the two that has a tuner and then they can use composite, component, or HDMI to get the signal from the device to the TV. Again, it doesn't matter with BHN. Their analog signal will work for some time and she (the OP) can use the cheap VCR to tune the TV if it doesn't have a tuner.
Think about it this way. It is obviously more expensive still to make a digital tuner in the TV. It is obviously stupid to encourage peole by adding an old analog tuner to a new HDTV. So these low end TVs have gone the route of assuming you will use an external tuner. Should we stop calling this a TV?
You know of course I am playing devil's advocate with you. Chances are there is a coaxail input somewhere on that TV that accepts an analog OTA antenna and has a tuner. My only concern with what you said is the notion that the store mis-labeled a whole row of devices as TVs and not monitors. That seems highly unlikely that they would make that big of mistake. I know some store workers are idiots but they have managers and QT teams to keep things straight. I don't think she was shopping for TVs at Big Lots, where the stuff is all just thrown against a wall in no order (price, size, brand, etc).
But anyway...... let's get something straight. It has already been gotten straight but I am adding to it. If you have cable, satellite, or vcr, dvd, dvd-recorder, etc... these devices will still work after the digital transition. The only thing that will not work will be the OTA antennas on an analog set. That is why they have those stupid converter boxes.
Obviously the converter box has its own tuner for the digital signal and then it sends it out as analog to the TV. Has anybody actually played with these devices? I have seen them at Target. I'll bet they don't just have a coaxial output to the TV. I'll bet they also have a composite output to the TV. So in the sense of this setup not using an internal tuner, you could call the display a monitor. Maybe all these new screeens are going just composite, as not to confuse the consumer. Also, keep in mind, most TVs for at least the last 15 years had composite inputs... whether it be just mono sound and video or left and right audio channels with video.
Somebody said the cable companies are trying to confuse consumers. I agree with that. What better way to get those final holdouts to upgrade to digital cable, once and for all.
Get 'em with fear. It is the American way.