The weights aren't needed to make it reproducable. The code and training data are needed. Hopefully if you used those, you'd ultimately reach the same result.
I guess I'm saying that if there are reproducibility problems without the weights, then there's still a reproducibility problem with them. A paper with weights that magically work, when training on the same data and algorithm doesn't work is a paper that isn't reproducible.
IMO, having the weights available sometimes just papers over a deeper issue.