But do you care about how they disproved it? Or do you care about the outcome? The process is interesting, can provide useful information, and a good process is the only way forward we have, etc., but it's not the point of science. What good is the process if the outcome is bad?

This may be a linguistic thing, though. Perhaps we read that phrase in a different way. But to me, it very much echoes "it's about the journey, not the destination".

Kind of sad to see my point downvoted. How something is disproved is very important. It tells us our theory is wrong. When we see a theory is wrong we reject it - this is the crux of the scientific method. An experiment by its nature pushes a theory to its limits. Going back to the example of Michaelson Morley, the experiment confirms that ether does not exist. Yes, the outcome of the experiment is that the ether theory doesn't hold. Yes, special relativity explains the micelson morley experiment but the converse is not nessecarily true. Now it is possible that the standard model and general relativity are the absolute truth, buth it would be unscientific not to try to find better explanations.

Going back to your point about "do you care about how they disproved it?" Yes, I do, and you should too. The michaelson morley experiment was a properly performed scientific experiment with every step documented so someone could replucate it. It is a source of truth which I can trust. On the otherhand if a tom dick and harry claimed that they had a vision and that this vision told them relativity is not true, I would definitely not trust such a claim.

Your point about the result being important is valid. Yes, as an engineer or as a business man results are far more important than process. But a scientist, must care about the process more than the result. If they didnt we would not be able to trust their conclusions.

The process is important, but why is the process important? So you can reach an actual conclusion. Your point about proving vs. disproving has absolutely nothing to do with the statement in question. Both proving and disproving things are outcomes of science, not "the quest".