Einstein, in the famous EPR paradox paper[1] used quantum mechanics (then fairly new) to show that it predicted some truly unexpected behaviour that we now call entanglement, but which Einstein referred to as "spooky action at a distance" (This is the source of the word "spooky" in this context, which has been picked up on by new-age nuts). Einstein didn't think entanglement was real at this time, but that QM's prediction of entanglement showed that QM was broken or incomplete in some way. No method was then proposed of testing whether or not Einstein was right (it was likely beyond the technical capabilities of the time).
John Stewart Bell came up with with a theorem[1] that would be developed into experimental tests that showed what Einstein thought was an error in QM was actually observable. It's worth noting that closing all possible loopholes in experimental tests of Bell inequalities is difficult and elusive. The consensus is that QM is valid and entanglement does indeed exist, but there remains room for a tiny amount of doubt. Note: This is doubt the way scientists see it. i.e. A reason to perform further experiments. Theories can never be proven true, but teasing out every nuance and conceivable way in which they can be wrong, and testing them, is what makes theories accepted and useful. QM has stood up to enough tests that it's now a very well established theory. However, continuing to test QM is not useless. Searching for new ways in which well established theories can be wrong is how we found QM in the first place!
Note that QM does not say what, philosophically, entanglement is or explain the mechanism for how non-local correlations propagate. There are other theories that have attempted to explain this, but none have been successful so far in being both correct and, more importantly, testable.
There have been other inequalities experimentally tested in addition to Bell inequalities, such as Leggett inequalities[3]. To your understanding, these would not be interesting because they don't "teach us anything new". This is false. The question about QM and entanglement isn't as straight forward as a binary "yes it exists" vs "no it doesn't". There exist subtleties. Different inequalities have indeed enhanced our understanding of entanglement and proven useful in practical applications such as quantum computing and quantum cryptography. I haven't had time to dig into the work linked to here, but to dismiss it out of hand is not reasonable. The press release (to be taken with a grain of salt given the usual failings of science journalism) does suggest a novel approach that could be of practical use.
[1]https://en.wikipedia.org/wiki/EPR_paradox
[2]https://en.wikipedia.org/wiki/Bell%27s_theorem
[3]https://en.wikipedia.org/wiki/Leggett_inequality