It is difficult to read the authors' minds. I think back in 1975 it would not have been appreciated the extent to which binary interactions can influence the evolution of compact binary systems. So probably the authors thought that this was a neutron star formed from the very recent core collapse of a massive star (and they were probably right).
If that were the case then it seems unlikely that the companion could be a white dwarf, since the progenitor of a white dwarf would have been of lower mass and should have lived much longer than the combined lifetime of the pulsar and its massive progenitor.
Indeed, the pulsar you mention in your question is similar to that presented by Hulse & Taylor in terms of its spin period and orbital period. However the origins are probably different.
In PSRJ1141-6545 it is likely that the white dwarf companion formed first but transfered lots of mass to a companion that then became massive enough to undergo core collapse and form a neutron star (Davies et al. 2002). So the pulsar could still be "young" but its progenitor actually lived longer than that of the white dwarf companion. I guess this possibility was not considered by Hulse & Taylor.
In PSRJ1915+16 it is likely that one neutron star formed first then the massive secondary underwent Roche lobe over flow and the orbit shrank. But the core of the secondary continues to the core collapse stage and the result is a close neutron star binary.