How to propose new notation

There is no sure thing here. The history of mathematical notation is interesting and varied. They come and go. The usual places for advocating new notation are books, journals, magazine articles, letters to the editor, personal letters. Don't be surprised that people won't adopt your new notation.

For some background you might want to read the Florian Cajori book A History of Mathematical Notations, reprinted by Dover Publications.

Because of all the cranks out there, it is quite difficult for an outsider to convince the mathematical establishment of a fact. Look at the story of Kurt Heegner, the Havergal Brian of mathematics.

Thus to convince the mathematical establishment that a notation is better than what they' been using, that might as well be impossible, even if it actually is better (which it may or may not be).

In principle, mathematicians are willing to accept something is true if you can prove it. If you can prove the Riemann hypothesis is false, mathematicians will accept your discovery, even though potentially it could perhaps invalidate lots of work that depends on the Riemann hypothesis being true.

But if you're asking them to change the way something has been notated for decades if not centuries, without having a concomitant disproof of something widely believed to be true, you can just forget about it.

Take the Legendre symbol, for example. $$\left(\frac{a}{p}\right)$$ Looks like a fraction, doesn't it? Or maybe the guy meant to write a binomial coefficient?

But even though it's ambiguous notation, even though some mathematicians agree that it's ambiguous, even though you can't really type it on a typewriter, even though the TeX for it is kind of annoying, the notation for the Legendre symbol is here to stay.

It doesn't help the would-be inventor of a great new notation for the Legendre symbol that it's called "Legendre symbol" rather than "Legendre function." The terminology is just as entrenched as the notation.

Maybe you can come up with some super brilliantly simple but unambiguous new notation for it. But all the facts pertaining to the Legendre symbol have been proven, verified and published for a couple of centuries or so.

Adopting a better notation for the Legendre symbol would be only slightly more likely than getting everyone to speak Esperanto and use the duodecimal numeral system.

An example of such a thing is $\color{red}{\text{Feynman's trig notation}}$

In his book, $\mathbf{\text{Surely You're Joking Mr. Feynman}}$, he says

I thought my symbols were just as good, if not better, than the regular symbols--it doesn't make any difference what symbols you use--but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, "What the hell are those?" I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.