The best advice I heard is that if you absolutely have to be at a particular place, go there for residency because most spots will give internal candidates priority. If you just want to do something in a general region the majority of neurology fellowships are buyers markets (my PD tells me easier to get than residency spot). The competitive ones are the residents from other specialties might also apply to (interventional, sleep, etc).
As for finding jobs after fellowship, it's just like finding residency after med school. There's a huge shortage of clinical neurologists so finding a job outside of academic shouldn't be a problem (unless you're aiming for a saturated city like NY or SF). For the competitive/academic jobs, the name and regional or institution specific ties (ie the PD went to X/Y med school) matters to a degree but are not exclusion criteria.
Don't stress the rankings/tiers because everyone has different criteria. But I'd say going to an academic institution with access to research/protected time/global health time/etc (whatever you're specifically looking for) in order to tailor your resume is important. I wouldn't drop so low where the opportunities you need to succeed aren't available, but also wouldn't stress a top 5 program (partners) vs a top 25 program (Mt. Sinai) because at the end of the day employers care more about what you actually accomplish rather than what your institution accomplished. If you work better in a lower stress environment, going to a lesser ranked program that fits you specifically better may be a better career more in the long run.
Not sure what you mean by chill, but I don't think you can go anywhere, be on cruise control, and expect to be competitive with the competent people who worked their asses off.
