Where can I do that? Do I have to manually extract each citation from a paper and then individually look each up? Can I validate quoted text without a subscription? Is it the same system for every paper in every journal or do I have to have fluency in multiple databases.

Yeah. It’s trivial to do for one paper while sitting at a computer in a university library. How trivial is it to validate the hundreds of sources and quotes that appear in any given volume of a journal?

I'm sympathetic to the idea: we should have an open, publicly queryable citation graph. Google scholar could very easily offer this at marginal cost near zero, but they won't.