There's an interesting statistical effect here.
Let's say there are 50 Uber drivers. 10 of them are doing it full-time; the others are doing it 5 hours a week for some additional income. That's 10 * 40 + 40 * 5 = 600 hours of driving per week, of which 400 is by the full-time drivers. So if you ask your driver whether they are full-time, they will say "yes" with about 2/3 probability, but the number of part-time drivers greatly exceeds the number of full-time drivers.
This effect is even stronger if you typically take Uber at times when part-time drivers are less likely to drive.
To your last point, location does matter a great deal here. Anecdotally, in my suburban area, I know a fair number of stay-at-home parents who do some driving on the side, esp. while the kids are in school. They most emphatically do not want to be driving full-time, or doing anything else full-time: if they wanted a full-time job they would not be stay-at-home parents. I am quite certain that this dynamic does not hold across all Uber markets, by any means.