I suspect the problem is not rejections, but that people don't bother submitting them because it's not exciting.

Has anyone here tried hard to publish a negative result and had it consistently rejected?

Yes, that the researchers don't push to publish them is what is dishonest.

Is it really feasible to publish everything?

And if you do publish everything, then everything will be lost in a sea of noise... you'd need to create "ultra-publishing" to resolve that...

Yes you can. When the answer to a unsolved problem could be either positive or negative, you can always publish negative results.

What you can't publish are things that are not novel and things that are worst than existing methods.

the exception is when the deviation from expectation is significant ie, 5-sigma certainty particles not conforming to speed of light limit would get a pretty quick acceptance.