I think currently the main reason why not more factories produce the BionTech/Pfizer Vaccine is that it is quite difficult to manufacture and so it takes time to set up additional capacity. As far as I know several competitors are in discussion to build additional capacity.
Of course there is the question why this has not been done earlier.
It kinda reminds me of the current cutting edge chip shortage - eq. the special microfluidics machines needed for mRNA vaccines VS EUV machines needed for cutting edge ASIC manufacturing.
