Drunk people are stupid. A giant "get me home now" button is more likely to get utilized versus stumbling around trying to wave down a taxi.
Also the cash thing, knowing that a card is 100% going to be accepted is huge, compared with Taxis and "oops the card reader is broken today".
I prefer to pay cash, personally. Denying finance parasites their 2% fee is always worth an extra ten seconds of my time.
In Seattle, you can't really flag down a cab.
What you could do in the past (pre-uber) was call their phone number, have the dispatcher on the other end pick up and be annoyed that you are bothering him, and request a cab that may or may not show up.
Also you had to know where you are at, which for a lot of people I know, is a serious challenge. A surprisingly large % of people have serious problems finding what intersection they are at. (To be fair, sometimes you are on a street that, for whatever reason, doesn't have any street signs, or the street sign is not lit up at all and you have to use your phone's flashlight to try and read it... and again, all this while drunk).
The experience sucked.
This has been the complete polar opposite of my experience in NYC. The professional taxi drivers are pretty consistent, like 1/4 uber/lyft drivers barely know how to drive at all. I distinctly remember one driver so terrifyingly incompetent I made him pull over and let me out of the car five miles from where I was going...