used a developer solution for over a year in a plastic bottle full of air
No, not full of air! Actually, very carefully devoid of any air - so entirely full PET bottles. But it is true I've been using the same batch of developer with adequate replenishment for a long time. I'm still using that same developer as shown in the video - it's just that it has been replaced gradually many times over since then. It's a bit like "grandfather's axe".
I wonder if it's possible to buy a single RA-4 kit and keep replenishing it with own chemistry
Well, you could, but it would require that your 'own chemistry' is essentially a fully functional RA4 developer. And if you make that, you wouldn't even have to buy the initial RA4 kit (or, at least, the developer part). A replenisher typically very closely resembles the working product, but with some relatively minor modifications.
Btw, it's also quite challenging to make (at home) an RA4 developer that has the same performance, stability and keeping properties as the commercially available products from e.g. Fuji. One issue is that it requires a number of compounds that are difficult to obtain for a private individual, and/or that are only cost effective if purchased in quantities far larger than you'd ever use in a lifetime. Another is that there are no officially published formulas around of contemporary RA4 developers. I've done some rooting around and collecting bits and pieces on my blog here:
https://tinker.koraks.nl/photograph...lor-developer-formulas-including-c41-and-ra4/ However, this is all scraped from the web, and that in turn is often second-hand information, with the best qualification usually being "works quite well for me", or "is purported to be very close to the Kodak formula". Make of that what you will!
Whether purchasing a commercially made RA4 developer is a 'waste' - I wouldn't say that. In fact, it's very, very cost effective if used wisely. When all is said and done, I have spent a LOT less on printing color in terms of materials than on B&W, on a per-square-meter comparison. This is partly due to currently still low cost of paper, but also because the chemistry lasts a looooong time (and isn't too expensive to begin with).