There was of course a ton of testing at the component level, and iteration when necessary. This worked for Saturn V because it could be done for each component in parallel, due to the huge mid-sixties spike in funding. But at the level of the integrated system, most of the testing—and nearly all of the iteration—was cut out, since it would have to be done in series for the entire rocket (or its stages). Thus the design had to be set in stone before the vehicle ever flew. The first and second stages of Saturn V flew just twice before crewed flight began. This was the right process for Apollo’s unique set of requirements and resources, but it is a bad idea when you A) don’t have a hard deadline forcing you to forego system-level iteration and B) don’t have a spike in funding to accelerate the up-front development, which otherwise takes forever. Hence SLS.

Granted, I am biased towards propulsion, but we are not talking about some bolt or bearing as a 'single component'. The propulsion system is a complex thermodynamic cycle, with tanks, turbopumps, preburner, regenerative cooling, injection, engine, nozzle, etc. Seems to be a rather involved 'component'.

That's just one engine - much less than a 5-engine stage even on a test stand, not to say about flying.

And combustion instabilities they were fighting were rather novel at the time, the engineers basically had no other options than to test a lot of variants - and invent debugging techniques with "bombs in the chamber" along the way. Fortunately it worked - low pressure of F-1 helped to mitigate problems of size.

I don’t understand your first paragraph - is this related to the F-1 development having strong iterative components?

Also, Combustion instabilities date back to at least the A4/V2, not quite new, and largely the same engineers.

The F1 is a bigger (!!) version of the same engine. It's got the same crucial features:

1. turbo-pumps

2. the nozzle is cooled and the fuel warmed by making the nozzle out of tubes through which the fuel passed

3. holes drilled in the tubes so fuel leaked into the combustion chamber to provide boundary layer cooling

I think the pogo-ing was solved the same way, too - putting baffles in in various places.

By that logic essentially all cryogenic engines are “the same”.

In a similar vein, all modern jet engines can trace their core features back to the Ohain engine (not the Whittle engine).