Short answer: Yes you are right, but classic pianoforte notation is too common to be avoided, your best bet is to try to find either another notation that is also common like MIDI or Tablature, or either find a ressembling notation system that is easy to convert back and forth with the classical one like the Simplified Music Notation.
Long answer: You are totally right, one can use any notation one wants.
In linguistics and semiotics, a notation is a system of graphic or
symbols, characters and abbreviated expressions, used in artistic and
scientific disciplines to represent technical facts and quantities by
convention. Therefore, a notation is a collection of related symbols
that are each given an arbitrary meaning, created to facilitate
structured communication within a domain knowledge or field of study.
Thus there's no reason why you should not use your own notation system, as long as it can represent what you want to express. So why was the piano notation become so widespread?
As its name implies, it's because it's directly inspired by a piano rotated vertically (see the correspondances between the left and middle part):
(Courtesy of Ken Rushton aka MusicScienceGuy)
So as you can see, the primary reason for making this notation was not because of expressiveness or any other artistically elevated concept, but because of practicality: piano was the most used instrument historically in occident and this was a kinda straight way to represent a score to play on this instrument.
And this historical relationship directly influenced the notation a lot further: at first, pianos didn't have accidentals (the black keys) but only the white keys, and thus that's why they are not represented on the notation. Accidentals came later, and to accomodate this evolution, the notation was "tweaked" to be able to represent them. But it's hardly a perfect fit if you see what I mean...
Another historical anecdote: piano was initially made primarily to play in the C key (since C was and still is the most used key), hence why chords are so easy to play in C, but so weird and hard in other keys. Some attempts were made to fix that which are called isomorphic keyboards, with a common mapping being the Wicki-Hayden/Jammer used on concertinas, and an easy accessible mapping being the colored traditional keyboard.
About expressiveness, in fact, MIDI is the most expressive standardized notation, and is a superset of standard piano forte notation (and you can often find a "convert midi to notation" option in softwares, but the other way is much more hard because notation lacks informations that can be encoded in midi, so to do that softwares must rely on AI algorithms or just on the user to fix stuff).
But even with midi, after using it for a while, you will soon come to the conclusion that it's not nearly expressive enough to convey and reproduce all the expressions you may want. There are more expressive specifications in various softwares but these internal notations systems are not considered standard (since most are closed source anyway).
Other solutions exists, and a lot of alternative notations have been made over the centuries. You can find a good list with critical reviews at musicnotation.org and a historical review here.
On the simplicity side, one of the most common but simple notation is the tablature, which is quite widespread for guitar songs.
As you can see, you have a wide range of possibilities, and can easily imagine anything between tablature simplicity and midi expressiveness, or maybe even beyond or combining both.
Why then use such an old, deprecated musical notation system that can't even account elegantly for accidentals nor the latest findings in musical theory like microtonality?
Answer: because it's popular and culturally anchored.
Pianoforte notation is the defacto choice for teachers in musical schools. Furthermore, nearly all scores use this notation, thus if you want to use another notation, you will have the double burden to first learn the pianoforte notation and then learn how to convert it into your own notation of choice. But if you learn pianoforte, there's not much incentive to then convert to another notation...
In fact, that's not quite true. Since, as I demonstrated above, pianoforte is not the graal of expressiveness, a lot of experimental music composers and some contemporary classical composers use their own notation system, sometimes just twists on the classic pianoforte, others making a whole new notation system.
So in the end, it's up to you to choose the musical notation system you prefer, but you should not only choose it because you feel comfortable with it: it should also be expressive enough for your needs, and most important easily convertible back and forth with the classical piano notation, or sooner or later you will get tired of using your notation.