Yes, it is possible. There are organisations who act as third-party sponsors for J-1 visas. An example is http://cipusa.org/. These organisations help with some of the paperwork, lowering the amount of work required by the applicant (you) and the company.
You are probably right in that the amount of formalia and paperwork is rather large for a small startup, but there are certainly larger companies that are willing to put in the work. I did an internship at a company in the US, and I live in Sweden.
I urge you to try, an internship in another country can be a great learning experience.
Perhaps you can shed some light on how things work from the inside:
How much interest is there for foreign interns, what is the competition and expectations like? Were there many people like you?
In terms of compensation and cost, what is to be expected? From what I gather, one needs to pay for private health insurance. Is it reasonable to expect that the compensation from the company could cover the plane tickets, health insurance and perhaps accommodation to an extent?
You are probably right in that the amount of formalia and paperwork is rather large for a small startup, but there are certainly larger companies that are willing to put in the work. I did an internship at a company in the US, and I live in Sweden.
I urge you to try, an internship in another country can be a great learning experience.