The OP said "I get that this isn't a major issue, but DS was very upset. DH thinks I'm being ridiculous and this is a normal way to teach. Opinions?"
I think it is the putting the scores up that the OP especially resents, not the fact the child has to learn the times table.
And I do think this is a form of shaming. That's a strong word for it, yes, but shame could mean "...an unpleasant self-conscious emotion typically associated with a negative evaluation of the self, withdrawal motivations, and feelings of distress, exposure, mistrust, powerlessness, and worthlessness."
(That's Wikipedia for you).
Maybe a one off exposure to this may not be a massive thing but imagine being last a lot! Might that be unpleasant, make one feel self-conscious and have a negative evaluation of oneself?
My dd has dyslexia, is on the spectrum and has anxiety, not all kids react the same.
I regularly lost house points for my team due to my poor spelling. This negative 'stick' approach never made me better at spelling and only made me hate school. Luckily some things did inspire me and I went on to get a degree. I'm afraid for a lot of kids the stick doesn't work.
I think that the competitiveness is not always good. Between equally matched students, it is fine, but between all kids it is not.
If we want to give incentives to kids surely encouraging them to improve on their own scores, with achievable goals seems to be the way to go. Not comparing them to who is doing the best.
Imagine a child in a class with a bad leg, would school make them race against the other kids on Sports Day, then displayed their name in last place on the school board and think it will inspire them to run faster in future? I don't think a school would do that.