Why don't researchers request payment for refereeing?
Academics aren't upset about not getting paid for refereeing/reviewing - they're upset because journals charge too much.
There's really four points in the statement "academics do pretty much all the work for free and publishers get the money"
- Academics do most of the work
- Publishers do a comparatively small amount of work
- Publishers get the money
- Academics don't get any money
Just because people might object to some of the four points, that doesn't mean they object to all of them.
Academics do most of the work - Most academics wouldn't object to this state of the affairs. Of course academics do most of the work in publishing (especially reviewing) - they're the ones who are qualified to do it. You can't have some bureaucrat take care of reviewing the work, you need someone who knows the field.
Publishers do a comparatively small amount of work - Academics might grumble at this, but there's a comparatively little that the publishers are qualified to do. Typesetting, printing, maintaining the journal website, administration in the reviewing process ... and that's about it. All the actual content decisions have to be done by knowledgeable people (academics). There's certainly some journals which try to offload things like typesetting onto the authors, but in part that's financially driven ...
Publishers get the money - This is the main point of upset. Publishers charge what is viewed as an excessive amount. ... but it's not that they're charging money per se, it's more that academics lose access to the content due to expense. Most academics were completely satisfied when they had access through (paid) library subscriptions. It's only when budget cuts (and publisher price increases) caused libraries to cut subscriptions that academics got upset. But again, it's less having to pay for things and more not being able to access everything they need.
Academics don't get any money - This is the point you're addressing. However, I'd say most academics don't have a problem with it. Refereeing for a journal is considered by most to be community service - it's something that needs to happen, and they're the only ones qualified to do it. It's quid-pro-quo: others review your articles, and you review other people's. Attempting to make it a paid-for enterprise makes the person asking for the money seem greedy.
So where does that leave you? Academics (mostly) don't have a problem doing most of the work - and doing it for free. The complaints are on the publisher's side: they're charging too much for what little they do. Demanding that publishers pay academics for reviewing isn't going to change that. If anything, it will make journal access more expensive, as they now need to pay for reviewers.
So charging a fee when reviewing isn't going to fix the problem, and as Mark Meckes mentions in his answer, asking for one is only going to make you look naive and greedy to the (academic) editor who you're in contact with. Note that there's a very big difference between "Decline to review based on principles" and "Decline to review ... unless you pay me".
Leaving aside arguments for and against the current system, here's what will happen if you --- as an individual academic --- are contacted by a publisher and attempt to charge a fee:
You will most likely be contacted, not by a "publisher" per se, but by an editor, who is another hard-working academic getting little or no monetary compensation for their job.
You say something like, "This is my fee for refereeing an article."
The editor responds, "Sorry, we have no budget to pay for refereeing," and goes to look for another referee.
In the best case scenario, the editor --- who is most likely a well-regarded senior researcher in your field, the sort of person who may make or influence decisions over your hiring, promotion, grants, and the publication of your own papers --- now has the impression that you're deplorably ignorant about how the field operates. In the worst case, the editor thinks you're a jerk who's actively seeking to make life harder for other people working in the field.
In some fields, for some journals, they are paid a fee. [File this as yet another example of "academia varies more than you think"...] This approach seems to be most popular with economics journals - as they are heavily into studying the way people can be motivated by money, I suppose this makes sense!
An exceptional example is the Journal of Financial Economics - most articles have a single reviewer, who is paid $275 cash (and a ~$200 discount off their next submission), contingent on a timely review. Lower down the scale, the Journal of Banking and Finance talks about offering "tokens of appreciation" for reviewers; there's no cash value given but the phrasing suggests it's probably a good bit lower.
Now, why do these journals do it? Probably because they always have done, an explanation which applies to a lot of strange quirks of the academic system. But is it a good idea? The recent proposal by Scientific Reports to have a two-track paid- and unpaid-peer-review system was incredibly contentious, after all...
The Journal of Public Economics recently tested the system out - they took 1500 review requests for their papers, and divided them into four groups:
- a six-week deadline, but no penalty for missing it
- a four-week deadline, but no penalty for missing it
- a four-week deadline, and a promise of $100 payment for meeting the deadline
- a six-week deadline, but reviewers told turnaround times would be made public
All variants worked well. The group with a four-week deadline had an average turnaround time of twelve days less than the six-week deadline group. Payments took another eight days off the turnaround time, and "public credit" another 2.5 days. The money/credit groups wrote slightly shorter reports, presumably as they were more motivated to make a hard deadline, but the editors did not see them as of noticeably lower quality.
So... yes, payment can work. But it relies on the journal having the money (two reviewers is $200/paper), and - intriguingly - it's not quite as effective as the no-money-needed option of just giving a shorter deadline. And even when it does work, it only makes sense if done systematically by the journal. Individuals asking to negotiate their own review payments is unlikely to work in the same way.