Am I correct in thinking that previous rocket designs don't have any redundancy built in? This seems like a big improvement in reliability but I'm not super familiar with other rocket designs.
The answer depends on specifics. The NASA Space Shuttle was able to reach orbit after the failure of one of its three engines, but only if the payload and/or altitude weren't near their range extremes. In other circumstances, the timing of the failure might determine whether the mission could proceed.
When I worked on the Shuttle, one design guideline was that no single-point failure modes should be allowed if it was possible to avoid them. Obviously this guideline was frequently not met. A one-word summary describing the avoidance of single-point failure modes is "redundancy".
Historically, most rocket designs push the performance envelope so hard they have little or no margin. Much of this attitude is historical, government rockets mostly being descended from ICBMs. The other part is that the rocket equation severely penalizes extra weight, and the window between "robust" and "too heavy to fly" isn't all that large.