"While it remains possible that the spirochetes detected in xenodiagnostic ticks could be unrelated to the infection, we view this as unlikely. First, we used a highly-specific (OspA) monoclonal antibody for detection, which should only target B. burgdorferi. Second, our ticks are wild caught at the adult stage but lab-reared from egg mass to nymphs, so with the absence of transovarial transmission, any occurrence (very rare in Louisiana) in adults would not be passed on to progeny. Finally, we fed a portion of our nymphal ticks on clean mice as controls to validate the absence of spirochetes in our tick colony."
It's not the tissue that is fed to the tick. The ticks feed on the live apes themselves to simulate natural zoonotic cycles. The point you raise is interesting because without the availability of robust detection methods how can one ever be sure a tick is sterile? Never the less some precautions listed above were made to lower the risk of contamination. Basically they took ticks from a place that has a low prevalence of Lyme disease, raised them in a controlled environment, took the progeny which are even less likely to be infected, and then confirmed their lack of infectivity by letting them feed on non-infected mice as well as staining with OspA antibody. I don't see this method as a 100% guarantee, but what can you really do. This is just one of the challenges of this illness.