So are you, dear reader, for intellectual property laws or not. I see a lot of "anti-this" sentiment confused with "anti-AI [data collection]." The consistent view is one regarding IP in general - not reverse engineering your morals based on the case.
I do take your point, that someone against using twitter for training data in AI who then uses a service that might be violating copyright seems hypocritical. But first of all, the people voicing these opinions may not be the same person, despite commenting on the same website.
But more directly addressing your point, I do think you could easily find a consistent moral stance that thinks taking tweets to train AI is worse than republishing those same tweets on a website. An argument can be made that the person writing those tweets wants to be heard, so republishing the tweet is in line with (your best guess of) what they want. Whereas consumption of the tweet for AI training is less likely to be consistent with the wants of the original author.