Most journals now have a "supplementary content" section where authors upload their results. Anyone can download that without a subscription. Obviously, older results are still locked up in scanned PDFs that you can only download with a subscription, but your database idea doesn't solve that, either. As for making something mandatory... well, that's probably the best way to get scientists not to do something.
Practically speaking, papers probably won't look substantially different under your proposal than they do now, except that there is the extra step of uploading data to the "methods and data" database. Separating the body of the paper from the results and methods would be a pain in the ass to read, which means that you'll need to continue to include those things if you want anyone to cite your papers. As a result, I can't see anyone doing anything more than having an undergrad copy and paste the relevant sections into the database.
Frankly, I think you'd get better results by just declaring that the publishers have to give non-institutional readers free access.
Practically speaking, papers probably won't look substantially different under your proposal than they do now, except that there is the extra step of uploading data to the "methods and data" database. Separating the body of the paper from the results and methods would be a pain in the ass to read, which means that you'll need to continue to include those things if you want anyone to cite your papers. As a result, I can't see anyone doing anything more than having an undergrad copy and paste the relevant sections into the database.
Frankly, I think you'd get better results by just declaring that the publishers have to give non-institutional readers free access.