Unfortunately, if your hosting provider does not support providing your own certificates or somehow prevents the domain ownership challenge from passing, there’s nothing that Let’s Encrypt can do. It’s simply not technically possible.
Let’s Encrypt removed the financial barrier to getting a publicly-trusted certificate. It’s still up to hosting companies and website owners to make use of that. What Let’s Encrypt can do is make this process both as secure and simple as possible (which, of course, usually involves trade-offs in one direction or the other).
Ultimately, it’ll be up to the market to solve this. Browsers have already started showing warnings for sites that do not use HTTPS if they ask for sensitive information. At some point, all sites that do not use HTTPS will show such warnings. All that is increasing the pressure on hosting companies to simply provide HTTPS as part of their regular hosting packages (which many of them have started doing already), and at some point site owners will just stop using providers which refuse to change. This might not help you in your situation today, but it will help somewhere down the line.