but I'm wondering if I will see either a difference in quality if I were to use something like a Lecia Sofort, or one of the (newish?) Fujifilm instax?
There's actually quite a bit of difference between instant cameras and those that shoot film.
Film, the negative, is designed to be an intermediary process. It's designed knowing full well that someone will be using that negative to create a print that is much, much larger than the negative. As such, lenses are designed to resolve detail at an incredible level and negative emulsions can capture a range of brightness values. The negative itself is also, generally speaking, a thing that we try to archive and to keep it around for a good long time.
The Instax print, on the other hand, is not designed to be enlarged later. It's designed for instant feedback. It has less range than a negative, which will show as higher contrast with less detail in the shadows and highlights. Because of their small size and non-need to blow them up, lens resolutions are not prioritized. Many Instax cameras use plastic lenses - which provide a soft looking image. I'm using Instax for a project of mine, but it's not something I'd recommend you use to scan, as the quality just isn't that good to begin with. Instax are for quick checking of light/style and for remembering a time - not for scanning and blowing up on a computer.
I just feel like I would benefit more from being able to have instant feedback of a photo. So if we're filming a trick while skating, I can snap a photo and know for sure whether I got it or not, and not rely on a luck.
Instax are great for instant feedback, but so too is digital. I get it, you've found a look that you really like and you want to stick with it. (I'm using Instax for a baby book and I convert digital images into transparencies for making cyanotypes with them. Trust me, I get it!) But, I want to strongly encourage you to get a used DSLR. Here's why:
You're photographing skaters. I did just this thing back in High school. One skater to another, SLR's FTMFW!. You need the ability to use decently high shutter speeds to get stop motion on a fast moving board. Or, you need the control to drop the shutter speed to induce motion blur. With an SLR, you'll have this control.
Part of me really wants to point you towards a Canon 1V, as they're the flagship 35mm SLR of the day and cheap to get now. But alas, I think you'll be just better off with a used Rebel or 40/50/60/70D.
As for your comment on getting no useable shots...if you want to be a good photographer, you need to learn a thing or two about shooting. How to meter a scene, what ISO is, what the Exposure Triangle is, and what the limits of your gear are. Frankly, you should know whether or not you've got quality in the camera, even with a disposable. If you don't, it's because you haven't actually learned how to use the camera...yet!
As far as no usable shots, some might have to do with ISO or aperture, some might do with light hitting the film by accident, while others are due to trying to time someone doing a trick
No disrespect intended by my comment above. It sounds like you're really taking this seriously. As such, I cannot more heavily recommend the SLR for what you're doing (over an Instax). And if you insist on sticking with film, Do lookup the different offerings from Kodak and Fuji and look up how those films react with Pushing and Pulling and even Cross-Processing. I think you'll find these processes fun and experimental.