Does gold open-access with article processing charge (APC) really help access to science and save taxpayer money?
I also think that people can go to the library to get access to research.


You're assuming that libraries can pay for access. That's not the case anymore. Even Harvard univerity, one of the richest in the world, can't pay for all the journals its researchers need. I think none of the Universities I know have access to all the journals it needs. So, you can imagine a public library won't grant access to all the existing literature.

That's even worse in developing/not too rich countries. In this case, you can notice that most Open Access (OA) journals adapt the cost of publication to the wealth of the country the article comes from (for example Plos).

It can also be a problem for small enterprises, that aren't very rich, so they can't subscribe to journals and have to pay "per view", but need access to the latest research in order to innovate. So, non-OA journals are an impediment to the technological progress too.

And I will also add the fact that, even in rich countries, it is not always that easy to go to a library. For example, when answering here on Stack Exchange, I try to add links to research articles which can be more precise than my own answer. If the OP is really very interested in a complete understanding of the answer, he could go to a library. But in most cases, if he doesn't have access to an article through the Internet, it will just waste an opportunity for him to learn.


  I published in both OA and subscription-based, and I will gladly submit to OA journals if they end up being the highest quality ones in my field.


However, you're pointing to a real problem here. If the "best" journals are not OA, do you have to compromise your career (or your students') to publish in an OA journal? In fact, some people would answer that the Impact Factor-based ranking of journals doesn't make much sense (see for example this article). And it is one of the reasons for the creation of Plos One, a "mega-journal" accepting articles only based on their scientific value, and not on an estimate of the interest its conclusions might have in the future. But for sure, this is a hot topic.

And another real problem here is money. If the Universities have to pay both for keeping access to non-OA journals, and for publishing in OA journals at the same time, it will be even more expensive. No university can afford it. A proposed solution is green Open-Access, where the articles are just put in repositories, and nobody needs to pay neither for publishing, nor for accessing. With a good post-publication peer-review system, this could work. But it also implies a huge paradigm shift, with new problems.


  Isn't it reasonable to use the options that we have to freely give access to our work (self-archiving, sending preprint to people who ask politely, etc.).


That is in fact kind of green OA. But, depending on the license you agree with when you publish in a journal, that's not always possible.