I found that Google works well. It's rare that I find an article I want to read and can't find somewhere -- maybe on the author's web page, maybe copied to some public directory, maybe someplace else.

For you, perhaps. But for me... Well, I host 580 PDFs on gwern.net because they are not otherwise available publicly, and I link to >865 external PDFs (37 of which are Internet Archive or Dropbox, and would not be indexed in Google). So that's easily a third of the articles which I use somewhere, I cannot simply find it online easily.

I agree, papers are often publicly available somewhere indexed by Google, but I think that happens for less than half the papers I access.

If everything else fails and you really need it, the authors are usually happy to email you a copy upon request.

That's a good point! However, authors are sometimes slow to respond, and most authors die (or, less drastically, some lose copies of and access to old papers).