Updated 7/23 after OP's hints, and a look at the suggested wiki page
Answer:
TYPEWRITER
I didn't write this with the answer
OP (@Soha Farhin Pine) wrote this on a computer, and thus not with a typewriter
When I use my keys,
they make such a sound.
Typewriters' keys are known for making a distinctive, loud noise when used.
Many a bygone writer and poet
have dated me for their work.
Before the invention of the computer, the typewriter was the favored way for writers and poets (among other literary artists) to record their work.
In the olden days, I fared well
In those days, the typewriter was very popular...
The ones who play me have the same name, but they soon fell
when the treacherous device with spiders had word.
... and those who had experience with them ("same name" indicates that we're talking about typists, who are apparently also known as typewriters) would have needed to learn a new skill when the computer (which can access the web, hence "spiders") came to be. "Had word" likely refers to Microsoft Office Word, a favored text editor.
Talk I will of my forefather
whose birth was on the 16th
France’s ram was his father of Italy.
According to OP, 16 refers to the 16th century; from the Wikipedia link I learned that Francesco Rampazetto (an Italian) invented a machine to impress letters in papers (so, a forefather to the typewriter) in 1575.
A scroll — scan the diem, take the rite,
but don't read the magic letters. Then cut the large end from
the mountains where two directions meet, adding everything before.
Rampazetto's machine was known as a scrittura tattile - which can be assembled from SCRIT ("scroll") plus URA (URAL mountains without L) as well as TAT (first half of tattoo) plus TILE.
As OP pointed out, scrittura can also be formed from Scan+Diem=Scandium (SC) plus RITE minus magic letter (which seems to be E) = SCRIT.
My kind with a wheel of the name
two places before my finger in ring has
was a grapher who knew nothing of maths,
According to the wikipedia page, America's first "typewriter" was called a typographer, hence the "grapher" clue (yet not at all related to the type of graphs one uses in mathematics, hence "knew nothing of math"). The typographer was known as an index-wheel machine, which explains the first two lines of this set (as the index finger comes two places before the ring finger on one's hand).
therefore making a mistake at the very beginning,
Typo- is the beginning of the word.
which Burt burted out.
Will I am, but it's not my name
rather one of my ancestor's father's.
William Burt (according to Wikipedia) is the inventor ("father") of the typographer (the typewriter's "ancestor").
I’m made of
The word typewriter itself is composed of:
the action of using the man-made
type, what one does on the (man-made) device,
and then its natural counterpart, to express
write, a counterpart to type, and a form of expression
At last, a hesitating suffix.
-er. Which when put together, of course, gives typewriter.
