Not necessarily, thanks to programs previously established many rural areas (depends on the state partly i think) have things to assist seniors. My state, New Mexico has a state tax, but seniors get an generous credit, some of it for property taxes paid. i've gotten state refund even in years when SSA my only income.
We have a Senior Center in our village (pop: 800) that serves some of outlying farm/ranch folks too. The Gov Commodities food distribution is handled thru it. We even have a dialysis center. Property taxes go for equipment for Volunteer Fire Dept too. No jail here but we have a small police force and ambulance as well non-emergency Medical Transport thru Senior Center and private or government grant funded agencies.
We also have a clinic run by Presbyterian Medical, and are only a 1 to 1.5 hr drive (or bus ride--yeah there's a bus to city--only once a day each way, but it is there)from the University's Medical center (actually had my sleep study for CPAP thru them).
NM is ranked as one of poorer states in USA, but i think that's in part because the income figures from the Reservations bring down statewide median income. Thing is our taxes at work are pretty visible here (at least if you live rural) they help fund the services some rural places in other states may not have. And we have some of the most efficient road maintenance crews i've ever seen and i've lived a lot places.
Of course i have an advantage over some in that my daughter is my 'housemate'. Works full time (currently working a lot of OT) and shares expenses.