Just my $0.02. I used to work in Dripping Springs and would sometimes take 360 north to my house in North Austin. I've found that despite the congestion on 360 the drive is more scenic and less congested than going through Oak Hill and Mopac. But I really think that if they upgrade 360 they should include dedicated bike lanes on the shoulder and better signage to alert drivers about the presence of bikers. It is probably the most challenging ride in the city and also the most dangerous. About the only safe time to bike ride there is at 6:00 am on Sunday.
Now if you want to talk about a screwed up road, let's talk about the Y. What were they thinking when they designed that !?!?!?!?!