The mountain states are car states. Denver is not a city like New York where people go without getting their license until they are in their mid-20s.
Having Uber or great public transportation will not change the fact that much of the draw of Colorado is that you can drive two hours and end up in the mountains or on the plains. Colorado self-selects for cars and they will not be going away any time soon.
That said, once a car isn't needed for day-to-day life in multiple neighborhoods in a city, and likewise once relaxed parking space requirements make it such that keeping a rarely-used car isn't trivially easy, (and especially an economy where reliable used cars have become far more valuable and unnecessary line-items in the budget might be quickly removed) the economics for young adults tilt pretty heavily toward tossing in gas money to with your car-owning friends or renting a car for $100 every few weekends when desiring a trip to the mountains. And urban families might become one-used-car-kept-for-10+-years families as opposed to one-car-per-adult families.
My understanding is also that Denver has a pretty-decent network of bike trails and a relatively-mild climate, which also means two-to-five mile trips can be pretty easily covered without a car.
Of course Denver won't become New York— but it stands a chance of becoming less of an auto-required city and more like the DC area, which is a relatively young metro area for the East Coast with a mass transit system from the modern era and new urban nodes at outer transit stops, and many people still own cars but plenty don't, especially in the urban core.
Let me offer a concurring opinion - Uber seems incredible expensive for two or three trips a week, from my experience. It works best when you're with a group of friends/family members and you are all drinking or don't feel like dealing with driving and parking. I would never consider it as an option like a normal cab for when I just need to do the week's grocery shopping or run out to get something at midnight, when the light rail might have shut down for the day. The only city where I ever used it was Denver, actually. Is this different from others' experiences?
As a side note - can't you drive two hours to a mountain from most East Coast cities? The Catskills are only 2.5 hours from NYC. I thought mountains were much more accessible in Denver than a two-hour car ride.
The mtns in CO are certainly closer to Denver than 2 hrs, but most people are going deeper into the mtns (e.g. Summit county/Vail/etc.). You can get up to 8500ft in 25min from Denver if you want. IMO, the further one gets from Denver, the nicer it is (I live about 8hrs from Denver in the mtns of CO, and have lived in the state for 20 yrs.).
Uber now includes Uber X, "rideshare" type service with drivers in their own personal cars. Where I am, a 2 mile cab ride to the grocery store will be $12 without tip, the same on Uber X is ~$6. It's far cheaper and less hassle, and uber x actually shows up unlike cabs. The introduction of uber(x) has let me postpone buying a car for a few more years.
I agree with your statement, but "you can drive two hours and end up in the mountains or on the plains." - why would anyone drive two hours to be out on the plains?!
Having Uber or great public transportation will not change the fact that much of the draw of Colorado is that you can drive two hours and end up in the mountains or on the plains. Colorado self-selects for cars and they will not be going away any time soon.