Most people in the country have cars because they need to get to shops, work and entertainment. In general they have large carbon footprints unless they are isolated and poor.
You're assuming that country people drive more than city/suburban people do. I haven't found any data on that but based on where I live country people drive less. We tend to schedule our drive days and many of us leave home only one to four times a month.
We tend to heat with wood rather than using FF generated electricity, oil or propane for heating. We, I think, consume less. Most have gardens, many orchards. Some raise their own meat and don't used trucked in feed.
People living in the country are much more likely to work from home (or be retired) and don't do the daily commute or run out for a pizza type stuff.
a car driven lifestyle cannot fit within our remaining carbon budget
Let me fix that for you. A
fossil fuel fueled car driven lifestyle cannot fit within our remaining carbon budget.
And gas may be as bad as coal
Only if you-
1) Assume leaks will not be contained. (Most NG leakage is residential/commercial distribution. Not gas wells and shipment to power plants.)
2) Overlook the methane release by coal mining and processing.
A study by the National Renewable Energy Laboratory estimated that surface mined coal releases 1.91 grams of methane per kilogram and underground mined coal releases 4.23 grams of methane per kilogram.
Pamela L. Spath et al, "Life Cycle Analysis of Coal-Fired Power Production," National Renewable Energy Laboratory, June 1999
3) Assume a 1:1 replacement of coal with NG. What is happening, since NG is highly dispatchable, is that when solar and wind are available NG plants shut down. We're working our way toward a (guessing numbers here) 40% solar, 40% wind and 20% NG grid. Except that wind/solar + storage is starting to eat into the 20% NG.