Flying into Copperstate in the SuperCub a few weeks ago Falcon tower had me flying 360s because I had traffic faster overtaking me. Like some how I was going to somehow run down that Velocity XLRG that went screaming past me at 200+ kts once we got into the pattern.
FWIW, I do find the continuous turn to be useful in slower planes like my SuperCub where I typically fly very tight patterns. In my other plane, I often times come screaming into the downwind at 150 - 170 kts. I use the turn to base to scrub off excess speed and the turn to final to scrub off the last of the energy, drop the flaps, and hit my approach speed after I roll out of the turn. So I like two turns in the faster plane. If the tower wants me to keep it in tight in the fast plane, it gets very busy. In the SuperCub, the tower (at a near-by airport) usually tries to clear me to land while I'm on downwind just so they can watch me make the turn abeam the numbers and make the first exit from the runway.
As a practical matter, I use both types of patterns at uncontrolled fields depending on which plane I'm flying, and have done so for many years.
One thing to remember is that turns in the pattern are designed to allow pilots to see other traffic that may be above, below, or crossing in front of them. My home airport has a very unusual approach since it's one way in and one way out (opposing traffic) with restricted airspace next to the airport. So they want traffic to fly a straight in 6 mile final. With no turns, it's easy to have planes stacked vertically that don't see each other, especially if one of them plugs the wrong freq into the radio. Remember the stacked Bonanzas landing a few years ago? Yeah, that was our airport. Those guys proved the point about turns for visibility in the pattern.
-Cub Builder