Just to be clear, I don't agree with sleep training, especially not at 3 weeks old, but neurological damage is not caused by a baby being upset. Sometimes they do cry it out, like when you're stuck in traffic on a motorway and can't get them out of the seat, or when you've tried everything and they won't be soothed anyway so you stick them in the buggy/sling and walk around until they fall asleep.
Nobody would seriously consider that as having caused neurological damage, it's temporary distress. So why is it different if you do it on purpose vs unavoidably? I wouldn't like to do that, because I don't like to think about my baby being in avoidable distress. But it's about the distress that I would prefer to avoid if possible/only cause distress for a very good reason (e.g. medical treatment). Not because of some fear that distress in itself causes long term harm.
And babies with reflux, or CMPA etc also cry a lot. I don't think people would say that causes neurological damage?
It's babies who are consistently left for hours with no food, no changing, no cuddles, no social contact at all who end up neurologically damaged from neglect - it's not to do with the crying itself, it's an overall pattern of lack of care and not being able to form attachments and not getting their needs met at all.
I don't like sleep training because it goes against things which are important to me, but I don't think we can really claim that it's causing neurological damage, in the context of an otherwise loving, responsive relationship.