If weights are copyrightable, then models should need to obtain permission before using copyrighted works to train them. You can't have it both ways.
On the other hand, Meta can have copyright over the model through 'copyright in compilation', which protects compiled works, regardless of the copyright of the underlying material.
So, I fear that it may be possible to have it both ways. But realistically, I think we'll only know for sure when this is fought out in court.
Disclaimer: again I am not a lawyer, so take this with a grain of salt.
My proclamation could be considered to be in terms of what ought to be, in order for society to be just and to prevent a disproportionate accumulation of power in ultra large corporations, which is detrimental to society.
Even if the base model is copyrightable (possibly a big if), there is a valid question of whether a new model which essentially optimised for something else, but used the base model as a computational shortcut to make it far cheaper to solve an optimisation problem, is still protected by the copyright holder of the base model.
Most of the barrier to creating large language models is the computational cost of training, not coming up with the training set data, so if fine-tuning gets around the copyright issues and allows for better FLOSS-licenced fine-tuned models, that would probably be a good thing (although maybe it will decrease the willingness of companies doing training to release models at all).