I should be clear that I don't think it's an absolute requirement for the gentry to keep a carriage AND horses. However, it does seem to be a mark of a family who is not struggling financially to keep at least a carriage (the Bennets, of course, have carriage-horses, but not dedicated ones). There are plenty of exceptions--as you noted, people in towns didn't need to have a carriage since they could get places on foot or, in Bath, hire a chair, and single gentlemen could ride everywhere. Single ladies would not be expected to maintain a full establishment and, provided they resided in a town, needn't keep a carriage or horses. It was also fairly respectable for a small family to have a carriage without horses, and hire the horses as needed, which I think is what the Hursts did, and makes perfect sense for a man of more fashion than fortune. I understand that only the wealthiest and snobbiest would take their own horses to London, since traveling with post horses was much faster (viz. Catherine Moreland's dismay at the slow speed of General Tilney's travel outweighing the grandeur of traveling with his own horses) and it was easy to hire job horses in Town.
It's clear, though, that a pretense to gentility was somewhat undermined for a family that had no carriage or horses--Mrs. Bennet mentions Mrs. Long coming in a hired hack as a significant problem, and Emma scolds Mr. Knightley for not coming "as a gentleman should," even though he's single.
It's clear, though, that a pretense to gentility was somewhat undermined for a family that had no carriage or horses--Mrs. Bennet mentions Mrs. Long coming in a hired hack as a significant problem, and Emma scolds Mr. Knightley for not coming "as a gentleman should," even though he's single.