Why do the natives of the English language not use a definite article before "silent mode" while saying, "My cell-phone is on silent mode."?
Which "natives of the English language" do you have in mind? I ask this because you use the before the quoted noun phrase above. And when a native English writer/speaker uses the, we are making a definite reference to something. When we make a definite reference to something that is a plural noun phrase, we almost always have a specific group in mind, AND we usually expect that our hearers/readers know which group we are talking about. But here I have no idea which group of native speakers you are talking about.
If you use no article (also called the zero article) before "Native speakers of the English language", you are making an indefinite reference and do not have a specific group in mind; you just mean native speakers of the English language in general.
As far as
My cell-phone is on silent mode.
First, it would not matter if there were 10,000 available modes. You would still not use an article. Each mode is already unique.
This is similar to a location where there could be one million tables, and you would say:
My cell-phone is on Table 6852.
And we would not say the Table 6852 even though we have a definite table in mind. It is basically the name of a uniquely identifiable table. And silent mode can be considered a uniquely identifiable mode. Just like Stack Exchange uses User##### to uniquely identify users. We don't need the definite article before one item/user/table/mode that can be uniquely identified from others in a certain set of items/users/tables/modes.
As for the preposition, on is definitely the one to use, because it fits with already established uses of on.
You can shorten your sentence to
My cell is on silent.
My cell is on vibrate.
My cell is on loud/ringer.
My TV is on mute (mode).
My speaker/volume is on low (mode).
The heater is on high (mode).
And you can instruct someone to put their phone on vibrate or their TV on mute.
Using in instead of on in the above sentences sounds bad.
Note that a person can also be on hold--not in hold--when waiting for a person to talk to.
Ultimately, it's just a collocation that doesn't require either the definite or indefinite article. We have lots of those, including
I'm on top of the world.
The shot is on target.
The actors look better on screen than face to face.
That football player is on fire; he's scored goals in seventeen straight games.
We have collocations with in (stand in line) and at (be at school) also, but this answer has been long enough.
Because it is short and unambiguous. It uses the same structure as expressing any other mode your phone would be in:
- My phone is on vibrate.
- My phone is on loud.
The premodifier (silent, vibrate or loud) already clearly state the type of mode your phone is in. The use of the article here is obsolete and would sound off simply because it is normally omitted when speaking about a phone's current mode.