If you are writing the essay for a class in gender studies, or if the teacher is an extreme feminist, I would say yes, go ahead and use synthetic gender-neutral pronouns. If the essay is about sexism, maybe.
Otherwise, no. Very few English speakers are familiar with any given proposed set of gender-neutral pronouns. There are dozens of such proposals out there. Many English speakers have heard of the idea, but no one proposal is widely used. When I read your question, I recalled that spivak pronouns were a proposed set of gender-neutral pronouns, but I didn't remember just what they were. (Grateful for the link, though I suppose a search for "spivak pronouns" would have turned them up fast enough.) And I'm someone interested enough in language to be active on two English language forums. I haven't taken a survey, but I'd be surprised if more than 5% of English speakers could tell you what "spivak pronouns" are or would recognize them if they saw them in text.
For most people, when they first encounter such a pronoun in your text, they will think it is a typo. I suppose after they see a few they will figure out that this is something you are doing deliberately. They will have a hard time looking it up anywhere. If someone doesn't know that the pronouns you are using are called "spivak pronouns", what would he look up? Looking up "e" with Google is not going to give useful results. I suppose they could figure out what the pronouns mean from reading the context. But at the least it will be distracting and at the worst very confusing.
This is, by the way, why I think it will be very hard for any such set of gender-neutral pronouns to become widely accepted. If you invent a new word for a new idea or invention, you can introduce that word in the context of discussing the new thing. If it's truly something totally new, people will be grasping for a word to identify it concisely rather than having to describe it every time they refer to it. Like when the cell phone was invented, you COULD say, "One of those new phones, you know, the kind that use radio waves to connect to the communications network rather than wires, and that you can carry around in your pocket". But it gets pretty tedious if every time you want to refer to the thing you have to give a 15 minute explanation. So somewhere along the line someone said "let's call them cell phones" and someone else said "let's call them mobile phones" and maybe a few other proposals, but those two have become popular. When those words were new, they only came up in the context of talking about these new devices, so it wasn't a big deal to explain them when you used them. People expected you to.
But with new pronouns ... we already have pronouns. Yes, there are some complaints about implicit sexism, but the words we have do the job. Many people see no need for new words at all, and most don't see a very large need. And pronouns are used all the time, in everything you read or write, over and over. These words don't come up in just one specific context: they come up literally EVERYWHERE. When I'm reading about a new idea or invention, I expect to have to learn some new words. When I'm reading about mundane or familiar subjects, I don't. It's annoying and distracting.
If you personally fervently believe that we should use such pronouns, I suppose you could try and see what reaction you get. If you want people to understand them, you pretty much have to include a paragraph at the beginning of your text stating that you are going to use these new pronouns and listing them. Which, again, becomes a distraction from whatever you are actually trying to say.