To introduce a new, non-gender pronoun into general English I suggest a number if factors are necessary.
Firstly, it must be distinctly English. Ze is good, but frankly, sounds too French. We need a decent pronoun which doesn't sound like it should be spoken by an amusing foreigner.
Second, it will need to be introduced by someone, respected by most people. Elton John springs to mind, but he's not quite enough. being so openly gay, such a word could quickly be the gay pronoun, (as Ms is usually associated with unusually masculine women) A US President would be good. (Think of that terrible term Brit, which was introduced by G Bush Sr, to placate the French).
But a good, very English word is needed to get it started.