It's sort of the inverse. If you're treated either way, then the vaccine is unnecessary. There are risks, even if rare, associated with vaccines. Basically, there's no value added and a very slight increase of risks.

Any unnecessary use of antibiotics should be avoided, as it can create an environment for the development of antibiotic-resistant bacteria.

Yes, but I believe the most common antibiotics prescribed for Lyme disease are doxycycline and monocycline, two antibiotics that are used by millions daily (on a somewhat long-term basis) already for the management of things like acne.

While certainly not ideal, with 8 million prescriptions in the US alone, I'm not sure an additional potentially unnecessary course of doxycycline is the worst thing in the world when compared to the possibility that the alternative is full blown Lyme disease.