TL;DR: it doesn't sound like you should jump ship.

Everyday I feel like I want to leave to start something of my own.

This is a normal impulse, but until you're getting a solid signal that you can start something of your own, and without taking onerous financial risk, don't take it too seriously. It's "freshman wants to drop out and write a novel" syndrome. Everyone goes through it, and we survive. You'd be weird if you didn't have this impulse on occasion.

Size up your current situation for whether you're getting the mentorship and resources and connections that'll make you be able to be a founder for real, and preferably without severe financial risk. If you are, great. (I'm guessing that you're not there yet, but a good startup will put you on the path.) If you're not, find a way to change your job as it currently is, in order to make it so. This may mean befriending people in upper management (a skill worth learning) or furtively re-focusing your efforts toward networking and career growth (also a skill worth learning) or something else.

If you're considered to be a valued contributor, then see if the founders will introduce you to investors on a casual, social basis. Yes, it's a political risk to ask, but this kind of political risk is insignificant in comparison to the risk involved in starting your own project.

I am always so exhausted after writing code for 8-9 hours a day that I definitely don't want to come home after commuting for 45 minutes (plus I like having somewhat of a social life).

If that 8-9 hours of work is advancing your career, it's not wasted. If it's not, then start slacking a bit (once you've established a solid reputation based on your first months, you can) and direct your focus toward building contacts and advancing your technical knowledge-- that is, work hard on the parts of the job (as you define it) that advance your career and slack on the stupid shit that doesn't.

I feel like I will enjoy it more.

That's "grass is greener" syndrome. If you're a natural entrepreneur (and it's OK if you're not; they're only ~1% of the population, and people who are not that can still be great entrepreneurs) then you'd enjoy bootstrapping, even though you'd have no money most of the time and most people (including almost all of the VC-funded entrepreneurs) would find it exhausting and miserable, and would rather give up some autonomy for a stable income. (The trait of the natural entrepreneur is an outlier stance on the rich-versus-king spectrum.) It's completely OK if you're not a natural entrepreneur; I'm certainly not. That just means that you're going to want to work for someone else and learn on their risk and dime, for at least 5-10 years, before you're confident enough to go out on your own.

If you're not a natural entrepreneur, though, you're not going to start a business unless you have funding already delivered (and, by the way, that's probably the right strategy for most). If you're going to try to raise a seed round with no connections, forget "enjoyment". The odds are against you, and fundraising if you're not part of the in-crowd (and it sounds like you're not, because that's 0.1% of the Valley, and it sounds like you're under 25, and you're in an employee role at a startup) is fucking miserable. People have been driven to suicide by the fundraising process. It is that fucking bad. Of course, if you run into a business co-founder who comes with the resources in-hand... that problem goes away. Then again, you're also likely to be #2 at best. It depends on you whether that's OK.

Plus, I feel I'm really only learning from myself at the startup.

You're ~22. Unless it's a shit-house, you're not the best and most experienced programmer there. Find someone you can learn from and start asking for his or her time, preferably in small amounts. See if he'll talk about his favorite subject over lunch. Internal networking is as important a skill as external networking. Take this opportunity to learn it. Mentors aren't "assigned" from the top, and finding "a mentor" or "a rabbi" is really rare. It's more sustainable and practical to learn a little bit from a larger number of people.

This is a great opportunity to learn things (such as internal networking, and how startups work) on someone else's risk and dime. But it's not going to be handed to you on a silver platter.

I should stay for 6 more months and get my small amount of equity to see what it will be worth. Even at an absurd outcome >1B the payout won't affect my life all that much.

Employee equity is, except for executives, usually a fucking joke. It's good that you learned that early on. Don't get bent out of shape about the fact. It's just worth recognizing the equity is almost never worth making career sacrifices over. If you have 20% of a business, you might favor the needs of the business over your own career goals and reputation. At 0.05%? Pfft. Think of it as a lottery ticket, and keep an eye on your own career objectives because, unless you're lucky and become the protege of someone important, no one else will.

That said... even if the equity isn't a reason to stay, your reputation is. Six-month stints look like absolute shit on your CV, and if you leave a "rapidly growing startup" to go out on your own but can't raise, it'll look like you were fired. The job-hopper stigma is fucking shitty, but it's real. Tech is more forgiving of job-hopping than, say, Wall Street or government, but leaving at 6 months just looks really bad. I'd say that, unless the job is horrible, that alone is a reason to tough it out for another year.

If it becomes clear that the startup is going to do nothing for your career, then drop the focus on your assigned stuff and put time into building the skills you want for your next gig ("powerslacking"). It's easier to act productive and engaged when powerslacking (another skill you might as well learn early on) then when just fucking around.

Any other things to consider? Any advice? I feel like other people have had similar thoughts or even experience leaving.

Yes. The time to ditch a startup is when you're no longer growing with the company. If you keep getting better projects and promotions, stick with it. (It's too early to tell, because you've only been there for 6 months.) It's when the company starts growing above you that you should bounce. Good startups promote internally, but most startups are crappy and pull their leadership from the outside (a social-climbing mentality) and when it gets into the state that you know you'll never be promoted because externals are going to get the best projects and roles, that's when you should jump. You're not going to have a good sense of whether your startup is in that state, yet; it'll be clearer in 6 months to a year, and that would be a good time to reevaluate. Even at that point, I'd say that you'd do better to get another job (external promotion) than try to wing it on your own... unless you can co-found with someone who can deliver Sequoia and Y Combinator on an idea.

Good luck!