As a science librarian who began work in the days before electronic journals, these branch collections were at that time essential to any research library. Their effectiveness was reduced because library administration tried to avoid staffing them, but in the even earlier days before photocopiers, which I have heard of but do not personally remember, even unstaffed collections were of enormous importance. In their absence, faculty make private collections for their own research group from their research funds, which, in addition to being inequitable, is even less cost-effective than branch libraries supported from the overhead on the grants.
Even now, until the time when all material is digitized, they are an enormous convenience. The best library service can compensate for this by technical means, such as immediate scanning and transmission from the storage facility. Such effective service, even if only on a M-F 9-5 basis, is quite expensive, and it is commonplace to find original promisers of access successively diminished. Once there are no material facilities it is very easy to lower the quality of virtual ones to nonexistence. Not just in libraries: whatever promises of increased overall service administrations anywhere in any field of activity can promise from cutbacks are notoriously unreliable.
If the money saved were used to improve alternate means of communication, such as by funding open access, the balance would be very favorable, but there is of course no direct linkage, and such a system change is beyond any single institution or country.
David Goodman DGG