Just found this: https://www.mdpi.com/2075-1729/12/9/1442/pdf

Which suggests that the host specificity in mammalian lice is due to less opportunity for host switching (relative to birds), and closer interaction with the immune system putting them into an evolutionary arms race which requires specialisation

"Birds, mammals, and their lice diversified over similar timescales, with major lineages diversifying around the K–Pg boundary [21], so it seems that there may be fundamental biological or ecological differences between the lice hosted by birds and mammals that could explain these differences. In general, birds have higher dispersal capabilities than mammals, and this dispersal may provide more opportunities for host-switching [33]. The sedentary and asocial nature of pocket gophers, for example, has been used to explain the high level of host-specificity and cospeciation in their lice [5]. Another major difference is that avian feather lice have little interaction with the host immune system because they mainly consume feathers, which are inert, while mammalian lice consume blood or sebum and directly interact with the host immune system [61,62]. This interaction in mammal lice may lead to an evolutionary arms race and more coadaptation between mammals and their lice, making host-switching more difficult."