I love cultural differences like this. I live in the suburbs of the the biggest city in a small country, Scotland. It would be actively weird and creepy to drive someone you're going on a date with around. And you know, alcohol.
I guess that would be going off topic so I will drop it..
Yeah, I definitely get what you're saying. I live in a city that has probably the best public transport in the world (Helsinki -- bigger than Edinburgh but smaller than Glasgow), and the thought of needing a car for dating is AMAZING. It just blows my mind that someone would even have that thought floating in their head.
I get that, if you live in a rural area, whether it's the U.S., Scotland, or Finland, you'll need a car for everything. But at that point you wouldn't be asking whether you need a car for dating either, would you? If you didn't have a car in a place like that, dating would be the least of your worries. I know somebody already brought this point up, but still.
As for the OP asking people out, always go for it. I get that it can be terrifying in the beginning, but there is literally no downside to doing so. Later in life, you will only regret not trying. I cannot emphasize this point enough.