Yes, you absolutely can create your own solution based on MapBox-created open-source technologies. You can for example host your own tiles and consume them using their web front-end (mapbox-gl.js) without using their service. I have a working solution and am exploring using this for our company in a serious way. You cannot however use Studio (you used to be able to use the Classic version, as other people mention) to style which makes things challenging.
So it does take work and effort, and it is much easier to just use their tools, but as you said there is a base price too high for many, and that also puts your solutions in their platform.
They They are vocal advocates of open-source at many levels of their org and I don't see signs of them changing, but this is an example of the kind of thing you have to watch when using open tech (consider mapzen's recent downfall).
Generating the tiles is possible through some of their many modules, or these days anything that can produce a pbf/mvt tiles which is now widely supported. Consuming the tiles with mapbox-gl.js is not free of challenges but is also straighforward once you start using them.
The absence of that free Studio product for styling is the biggest barrier to creating a solution quickly and leveraging the tech the way they do, but you can learn to style based on their guides. Again, there is a lot of effort and creativity involved but do-able for dedicated developers.