As it is mentioned in the comments, some of his papers are available, but you have to search. Of course, your remark about "belongs to mankind" concerns not only Tate
but almost all mathematicians. Their work is payed for by the society, essentially by taxpayers, and therefore it should be in open access. But publishers tend to appropriate it to make profit. 

Of course you cannot blame publishers, since they are mostly private enterprise and their primary goal is to make money for their investors, rather than benefit the mankind. So to the question "who is responsible for this situation" the answers probably are: a) mathematical community itself, which delegated publishing business to the big private corporations, and b) legislatures which make draconian copyright laws (to benefit the publishers). Nowadays, the situation changes somewhat because of the existence of arXiv. Mathematicians who understand that ther work "belongs to mankind" post all their work to the arXiv.

Unfortunately this does not apply to old works, which are simply appropriated by corporations. They used to be in the public domain (through university libraries, which used to be open to everyone), but now, with transition to digitalization, they become private property more and more.