A month after asking this question I randomly stumbled onto the type of solution which I was originally seeking -- open-market subscription based access to multiple journals and full-text article links from sources such as Google Scholar, PubMed, EconLit, etc.
While searching for full text access to an article on Manufactured Environmental Toxins in umbilical cords I noticed that one of the full text options was through a service called DeepDyve.
It claims to be the "Spotify of Academic articles" (Spotify is a popular Internet radio app that lets you download and play music at will if you subscribe). Here is a somewhat dated review from Ohio State's TechTip a la 2009. It's a $40/mo subscription plan for non-students like the institutional access you get within academia. I'm on a 2-week trial of it now.
Of course, I'm still going to continue to make use of many of the other good suggestions and I'm on the lookout for other services like this to select from.
USE WITH ADDITIONAL SOLUTIONS
In addition to the service I found, I'm taking advantage of several other solutions offered. Even with the subscription-based service there are many papers and journals to which I do not have access and the follow suggestions remain vital:
- Many papers are freely available on authors' websites, and pre-print servers (use search engine to find those).
- Write to the authors, asking for copies. Majority of academics are happy when their work is read, and will send you a copy.
- Your public library might subscribe to more than you suspect. Check it out.
- Many institutions grant library privileges to alumni, which might include remote access to the university's online subscriptions. You might get in touch with the librarian at your alma mater and ask if they offer such a thing. (In some cases you might be required to join the alumni association and pay dues, but this would probably be on the order of US$10-$100 per year.)
- Many universities open their libraries to the public. If you live near any university or college that has appropriate subscriptions, you may be able to just walk into their library, sit down at a computer, and download the articles you want. Then just put them on a USB drive, upload them to a cloud storage account, or email them to yourself. For older articles that aren't online, the library may have them in bound volumes; they may not let you check them out, but you can photocopy or scan any article you want.
LIMITATIONS OF THIS SOLUTION
The subscription-based service isn't a perfect solution. @J.Zimmerman points out that, unlike institutional access, you do not have the right to print or download papers. It's "read-only" access.
The selection of journals is quite large, but still limited. My feeling is that it directly provides access to about the same selection you'd have with most universities, but unlike universities there's no inter-library loan or other work-around for when you do not have access.
As I use this solution more over the course of the next few days I'll update this solution with further limitations and I'll better integrate it with the other useful solutions which have been posted. I will also take a suggestion from the comments to make this a Community Wiki solution.
Finally, I will also be on the look-out for any competing services like DeepDyve. Please update this solution if you know of any, so that we're not inadvertantly providing an advertisement for one arbitrary commercial service.