Honestly, it's not that straight forward, which is why I didn't mind leaving the manuscript in the end. I did much of the work bringing it to a workable manuscript, but I didn't originate the project.

I was approached by the corresponding author, who had originated and done most of the initial work. This was just a data dump, and he felt he didn't have the time or the ability to polish it into a manuscript. My contribution was to analyse the data, and write most of the manuscript. However, it became clear that the two other authors were pushing a narrative which I felt wasn't well supported by the data we had. We discussed two possible options: bring in an immunologist to determine if our results make sense, which I voted for, or stick to our guns. The second option would have also required submitting to a preprint, which they were against. The corresponding author felt the claims were justified, and decided that a more forceful (and prestigeous) first author was what was needed instead, and I agreed. Since my contribution wasn't all that much (about a week of my time?), and I wasn't convinced by the findings, I didn't mind losing the authorship. This sort of negotiating happens all the time in academia, so it's unsurprising.