Soyuz has launched 394 people and killed 4 of them. The most recent fatality was 50 years ago. The Shuttle launched 833 people and killed 14 of them, most recently in 2003 (8 years before the program ended). That gives Soyuz a fatality rate of 1.0% and the Shuttle a fatality rate of 1.7%. In other words: You're 70% more likely to die if you ride the Shuttle instead of Soyuz.
The more you look into the specifics, the safer Soyuz seems. The Soyuz deaths were early in the program while the Shuttle deaths happened when the program was mature. (Soyuz 11 was the last fatality and it was the 10th manned mission. STS-51-L was the 25th Shuttle mission and was considered to be safe enough for a civilian teacher to ride along. Oops.)
Soyuz's design is inherently safer. It has a launch escape system (which saved the crew of Soyuz 7K-ST No.16L).[1] The Shuttle did not. Soyuz's heat shield is much more robust than the Shuttle's. Soyuz's crew section is on top of the rocket, reducing the chance that any falling debris (or shrapnel caused by an explosion) will harm the crew module. Lastly, Soyuz uses liquid fueled rockets that can be shut off at any time. The Shuttle's solid boosters could not be throttled or shut off. All of these design decisions make for a simpler and safer vehicle. If I had to pick one, I'd ride Soyuz for sure.
Of course proving a simple design and sticking with it is safer! Not going to space and being content with remaining on the ground is safer still! Water is wet! The sky is blue!
We can all see what happened: the soviets stopped spending money on innovation and tried to re-frame their stagnation as operational brilliance. Sure, Soyuz has a reliability niche, and I'd rather ride on it too, but I'd rather fund efforts to advance space exploration, even if they wind up being a clusterfuck like the shuttle.
I think he is referring to the Soviet https://en.wikipedia.org/wiki/RD-270 full-flow staged combustion engine from the 60s.
The Americans had not been able to preserve the high-pressure hot oxygen from eating the preburner.
Soviet spaceflights were quite long compared to American - Salyut-6 and Salyut-7 both had more expeditions than Skylab, and that's before Mir, while Shuttles were limited to about a month in space per flight. Of course launch and landing are more dangerous phases of flight - Apollo-13 notwithstanding - so it makes harder to pinpoint an objective measure for safety.
So combining with your statistic the U.S. has a fatality rate of 4.4% versus Russia's 3.3%. Given the low N it's probably within margin of error.