I'm going to throw in my two cents worth here.
Most modern TV's now have the coax connector for digital-only signals. Meaning that you can make use of digital TV antennas without any other electronics necessary. However, in doing so it will never be able to pick up an analog signal again. But, there might be a trick you can do to get your Atari to work on that TV...that is if it has composite inputs (yellow, red, white). Find a cheap VCR. It can be mono or hi-fi doesn't matter. Connect the 2600 to the VCR and change the VCR channel to the channel your 2600 is set for (2 or 3 in most cases). Use RCA cables to connect the VCR's video and audio out signals to your TV's composite inputs. Video will be yellow and if you have only one audio out on the VCR then either the red or white input on the TV should become a mono channel meaning even if only one port is used sound will output to both speakers.
If you don't have a VCR they can be picked up rather cheap depending on your location. I'm not sure where you are from, and mean no disrespect or anything, I just don't know how much a VCR costs in other parts of the world. Get a good used one for this...and it doesn't necessarily have to play tapes. As long as it turns on and can receive and send video and audio signals you can use it as a de-modulator. Almost all VCRs from the 1980s and up have this built in.
On a side note: I have a ViewTV digital TV box that does quite a bit (I can play movies from the Internet Archives on this thing from a USB stick!). I wanted to see if it would also pick up a signal from an Atari console since the ViewTV can record as well. I was going to try to use it as a game capture device. But, alas, it wouldn't detect it because it has no way of detecting analog signals.
With that being said you might be better off trying to find a converter that will change your Atari's signal from analog to digital much like those boxes that changed analog antenna signals into digital ones when it was being forced to remove analog TV signals completely. One of those might actually work if those are still around.