It is not true that all open-access venues require authors to pay article processing charges (APCs) to publish with them. For instance, in my field, https://en.wikipedia.org/wiki/Logical_Methods_in_Computer_Sc... is a reputable journal which is free to read and free to publish; it is hosted by the Épisciences platform which is managed by a French public agency (CCSD). People usually use the name "diamond open access" to refer to such journals, which I believe are the best direction for scholarly publishing. By contrast, we talk of "gold open access" to refer to the situation where traditional publishers charge APCs: as you point out, this approach has several problems, in particular the fact that the APCs are often extortionate (e.g., $2000 for a 12-page PDF provided by the authors in a completely typeset and publishable form). However it's not correct to say that open access is expensive -- that's only the case if one stays with traditional publishers.
It's also important to keep in mind that "Think, Check, Submit" is an initiative backed by traditional publishers (of the "gold open access" kind), and I think they are making this a bigger problem than it actually is. In my area, everyone knows which conferences/journals are reputable, and everyone knows that all others are essentially scams (especially if they have APCs). I have never seen anyone make the mistake of publishing valuable material with a scam publisher.
This. Besides "diamond open access" some in the open access community try to establish the "fair open access principles" [1]. For a journal to comply to these principles means that they are owned by the scholarly community, free for authors and readers, authors retain copyright, and any other costs paid by the journal to a publisher are low and reasonable. We have started to put up a network for journals that follow these principles with the "Free Journal Network" [2] to promote and support each other.
Concerning other comments on funding, I wanted to add that some journals don't need much money (hosting is basically free, workload shared well by editorial board). On the other hand funding can and does come from universities and their libraries (money not spend on subscriptions), research institutions, museums, and donations (sometimes seen as volountary APCs by those who have the money from grants).
From what we've seen, the problem that Think, Check, Submit adresses is not necessarily a huge problem in countries that has a strong academic tradition. However, once you start looking at places where academia is just emerging, the difference can be huge.
Also, it's absolutely worth noting, like you point out, that different fields have different processes for accomplishing the same thing. It seems that younger academic fields are a bit more independent of the traditional structures. I hear that in Computer Science, conferences carry more importance than they do in some other fields, for example.
In other fields, there's a lot of inertia to deal with.
It's also important to keep in mind that "Think, Check, Submit" is an initiative backed by traditional publishers (of the "gold open access" kind), and I think they are making this a bigger problem than it actually is. In my area, everyone knows which conferences/journals are reputable, and everyone knows that all others are essentially scams (especially if they have APCs). I have never seen anyone make the mistake of publishing valuable material with a scam publisher.