I feel that the word "on" sometimes has meaning of "using".
For example, in the following sentence, "Join Us On the Radio Tonight"

You don't mean that "join us on the surface of the radio, tonight", do you?
At least to me, I don't think it is what it means.
In some dictionaries, they explain it as an idiom, i.e. => "on the radio"

Some more examples,
(i) She will walk foot.
(ii) I saw it BBC World, heard about it the BBC World Service and then read about it in the Guardian Weekly

Even though, I can not remember more examples right now, but I am guessing that the preposition "on" is sometimes used instead of the word "using" but in the same meaning as "using".
But it's only what I am guessing.

Could anyone please help me understand it?!

Thanks in advance.

It is an interesting observation on preposition use. You are correct, "using" and "on" are at times interchangeable. For example:

I did it on the computer.
I did it using the computer.

But be careful - they often are NOT interchangeable. For example:

I heard it on the radio.
I heard it using the radio.

The second sentence above implies that your use of the radio was unusual in some way. Why? Because we don't normally "use" a radio to hear something, just like we don't normally "use" a TV to watch a movie. We listen to the radio, watch TV, or listen to programs "on" the radio, and watch movies "on" TV.

Here's an example of what I mean:

I used a special radio to intercept the police communication.

The above implies that you did something unusual with a radio for a special purpose.
May I put in my two cents worth?

I think the original sense of the 'ON' for this usage is 'by some means of conveyance'.
(1) You can go either on foot or on horseback.
(2) We'll leave here on the last train.
(3) We can hear her voice on the wind.
Maybe the 'ON' of the last usage was taken as 'by some means of communication'.
(4) We can hear her song on the radio (waves).
(5) We can talk on telephone (weirs).

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I don't think that a native speaker thinks of "using" as the meaning of "on" in these idioms. And neither word is derived in any way from the other historically.
I do see your reasoning to a certain extent. "using" or "by means of" can certainly be substituted for "on" in many cases.
Interesting idea.

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victorycountryI feel that the word "on" sometimes has meaning of "using".
For example, in the following sentence, "Join Us On the Radio Tonight"
My observation...
First of all, let me start by saying that your post gave us an impression of someone who has a rather good command in English because you seemed to have the correct capitalization and puncutation at the right places. Having said that, I am a bit perplexed by the way you perceived the use of "on". You compared it to "using" which has absolutely has no connection. On - is a preposition that can take on different complexions according to the context in which it is used.
Whether it is about " joining someone on a radio show", " appearing on a TV program", or " She is coming on foot".
On -does not mean or equate to " using". If you persist with this perception, it will likely create havoc for you in the process because how would you explain these phrases: We watch movies on demand, I depend on my wife everday, and Please come on time.
You may find thousands of entries on the use of "on" online, but most of them are fixed expressions or idioms.