^^^^ This is the main reason to get a cart.To add to this, if a cart allows you to finish a round without pain, who cares what kind of looks you get?
I know I wouldn't be able to walk the day after a two round tournament if I had to carry my bag. Like the OP, I have lower back pain after and during rounds to the point that I can't sit during tournaments without the fear of pain when I get back up.
if you want a cheap look at good will or yard sales to find a jogging stroller with large wheels. Most guys around DFW say they find them for $20-50. You can leave them stock or mod them how you see fit. pushing them up hills is easy as long as it's not through uncleared rough. I find there are very few courses where I know I can't bring my bag boy quad cart when I play.