I gather the "big" promise of the mRNA vaccines is that they take somewhat less time to produce than conventional methods where the target protein (or virus) has to be grown first.

The current problem with them is that they have never been accepted* for medical use, so there is no industrial capacity to begin with, and the demand is for billions of units.

The deactivated or vector-based vaccines take longer to produce, but the worldwide capacity exists and can even be expanded. They are basically held back by the fact that vaccine production, of all kinds, is an ecosystem and the capacity to produce equipment and materials for any kind of vaccine production just is not there yet.

For example Pfizer managed to get priority in USA for it's vaccine production requirements by strong lobbying and being approved first.

I would be surprised if the companies producing high-end lab equipment and material need to grow, filter, purify the proteins and all the adjuvants, stabilizers and preservatives that go into vaccines would be that keen to build new capacity before they know if this is one-off boom. They rather wait and make good profit from demand exceeding supply.

* even now they have been accepted for emergency use only. The very second there is either a working treatment for Covid available or a vaccine properly accepted, they lose the license.