You can do it but there are several drawbacks and tradeoffs:

1. Power drain - the GPS function is a major power drain on a smartphone and may make it a problem for longer rides
2. Display issues above
3. Durability - if your bike falls over (which happens!!) do you really want to trash your $300 smartphone??

That said, I do use the strava app on my iPhone. I don't have the phone "on" while riding though, I just push the "start" button, turn the phone off, and put in my jersey pocket.

I think that if you want a "real time" computer it's probably better just to get a dedicated device. Using the smartphone is a major compromise.