Just because you invited your friend to your birthday party and paid for her meal doesn't mean you can tell people she'll give them a free ride home without consulting her first. What state of mind would someone even be in to do that?
Being the friend who has a car, or in my friend group being the only one that has a license because everyone else is afraid of driving, can be hard. People tend to expect you to drive them everywhere, which can be fine if we are going to the same place, but to always have to pick everyone up, even when they don't live close by, is way too much. Having a car doesn't transform you into a taxi or a cab. You are just someone with the ability to drive yourself to places. You can take other people with you, but you also don't have to and can simply drive by yourself if that's what you want.
I think that you can always ask for a ride from that friend who has a car, but you have to keep in mind that whether they accept or decline, both answers are fine. You can ask, but you cannot expect a positive answer, at least not always. To me, there is some etiquette for asking someone for a car ride. First of all, you need to be mindful of where you are planning for the other person to drop you off. If you live 20 minutes in the other direction, then don't expect them to drive you. Second, if they do agree to give you a ride, you cannot complain about anything. They can be the worst drivers, and you still shouldn't say anything. They are doing you a favor, not the other way around. Third, if they are doing a detour for you that is pretty far, you need to offer to pay for gas. They are doing way more than they should have. And fourth, always be thankful after it. The "driver" did something for you; you could have just called a car, but np, they gave you a ride. The least you can do is be thankful and not assume that it will keep happening. Always remember, they offered you a car ride because they are your friends and want to do something nice, not because they have to.
So, if someone offers to give you a car ride like in this story, don't act like an entitled brat like the birthday girl from this story did. Friendship isn't some pact that makes other people always do what you want. If they don't want to give free rides to other people in your party, then they don't have to. Having a car doesn't make you a taxi, so just be grateful if the person driving is kind enough to give you a ride at all. To expect more is to simply take advantage of something nice that your friend was doing, and that will never be something that's fine to do.