The solution isn't publishing negative results. The solution is publishing better results. The solution is holding people to a higher standard and insisting on reproducibility before accepting the findings as factual.
There are times when negative results pose new problems or offer new questions or methods. Those should be published but don't normally get you much praise. A blanket statement about publishing negative results is just going to get overwhelming information that nobody has time or interest in.
Frankly, there are more productive things that occupy a researcher's time. If you want to know what a researcher has tried and failed, then chances are you're a professional in the domain. You should already know how to contact them and you can ask them to share their journals and logs with you.
But, nobody is going to submit all their negative results. Nobody is going to index, catalogue, cross-reference, fund, and disseminate these massive negative results.
If you want to know all the things I've tried and failed (I am a retired scientist), it would take multiple volumes and far more time than I'm willng to devote. Prior to retirement, I had positive results to publish, more things to try, and to find a way to make it commercially viable.
Those acts are far more valuable than publishing things that got a negative result. Publishing those would take all my time and I'd never have gotten anything done.
The system works. The system was designed as it was for a reason. Yes, some negative results should be published but they are exceedingly rare. The problem is a lack of effort to reproduce prior to acceptance and low-quality peer review. The solution isn't to publish more stuff, it's to publish less and to ensure the quality is there. The solution probably should also include open publishing. Instead of locking it up in expensive journals, make it more open so that more people, even layman, have the chance to review and replicate it.
Speaking of which, I mention again that I lament the lack of emphasis, appreciation, and support for the citizen scientist. I'm a scientist. I'm nothing special. I'm wrong more times, by 09:00, than most people will be all day. Stop putting scientists on a pedestal or in ivory towers.
You really don't want our many, many volumes of negative results. Just because it's got a Ph.D. tacked onto the end of the author's name doesn't mean it is valuable.
Really, just uphold the standards that many seem to have forgotten. You're correct in that some negative results should be published, but they are few and far between. If anything, we have too much being published and a distinct lack of quality.
I will add this...
Journals only want a trimmed down paper. A journal may want you to trim your paper down to 10 pages, some want even shorter. Your research may be 1,000 to 2,000 pages but only a tiny bit gets published, or even submitted. That full research will have many of the negative results, as may the 'private' journal that the researcher worked from. If you want that information, you can probably get it emailed to you. The institution may have it in their archives, if the researcher is no longer available.
So, negative results are out there, but most don't really need to be published. Most researchers don't have time, resources, or inclination to publish negative results with dubious value. Sometimes a negative result is potentially valuable, as it introduces something novel. Those get published with some frequency, though rating it as novel and anticipating its ability to further the arts is easier said than done. Even still, those rarities are even more rarely deserving of accolades. There are no participation trophies in science. That sort of thinking is kind of how we got into this mess. Somewhere along the lines, someone should have said, "No, that's garbage. Don't publish it under the heading of science or with the prestige of this journal/institution attached."