In comparison, the Saturn V used only 5, they were just that much bigger. It's widely regarded that the 5 vs 30 was the difference as to why Saturn V was successful and N1 was not, as the Soviets simply had to roll too many dice in a situation where a failure resulted in complete mission loss.
It's a lesson in system engineering, particularly around failure durability and parallel vs serial critical subsystems.
In this sense R-7 is remarkable because it has a good overall success record (most launched rocket) and 32 chambers working at liftoff. Some of those chambers are rather small - about 3 metric tons of thrust, about as much as main chambers on British launcher Black Arrow - but still.
You're right, though, the R-7 is a fantastic achievement not only for the chambers, but also considering how many engines and pumps it has. All those spinning, moving parts are another point of failure and they're all necessary for success.
If Soviets wanted something modern, they had to reinvent it from scratch - or steal it. They were fairly good at both. Whether it was a computer, transistors or entire designs. The results were usually very simple designs where little could go wrong and that didn't need complicated technology - which wasn't available. E.g. the Soyuz capsule vs. both the Apollo and Space Shuttle craft.
I lived in a former Soviet bloc country and I can tell you that getting even silly components like LEDs required superhuman efforts. The few that were made domestically ended up in industrial uses (or exported to USSR) and importing anything from abroad was almost impossible. We had a black & white TV full of tubes until late 80s at home - color solid state TVs didn't become available until 1988 or so (and sucked big time).