Maybe you should ask yourself another question:
If the US was developing their rockets conservatively in the 1960s, would it win the race to the moon with the Soviet Union?
The U.S. was developing some of its main technology conservatively in the 1960s, particularly the vehicles central to the moon mission.
The Saturn V was a conservative and outdated design when they started building it. It was basically 1950s concepts.
So was the Apollo capsule itself. It was more of an enlarged Mercury than the Gemini was.
Some central characteristics of the Apollo spacecraft were solidified near the beginning of the lunar program, in the early 1960s. It was less of a 'pilot's spacecraft' than Gemini was. Gemini was seen as a stepping stone, and open to more experimentation, so it had a more efficient orbital module [SM?] than Mercury + Apollo and better cockpit. Because it was more allowed to take advantage of 1960s design concepts than Apollo was.
Apollo/Saturn was so big and expensive that once the design was largely settled upon, they had to work at it for a long time, getting those big engines ready for Saturn V and so on. Saturn V was a logistical nightmare. It required a standing army of personnel. And to prepare one for launch involved getting a lot of mechanisms ready that had hard limits on how fast it could be prepared.
Something like Soyuz was more revolutionary, more advanced in concept than Apollo. Soyuz only brought as much crewed [i.e. more expensive] weight as was needed into orbit, with the orbit-only components having lower weight as they didn't need re-entry shielding. So the only heavy, shielded part was the crew capsule, which detached from the other 2 parts. Soyuz had a higher ratio of usable crewed space to Apollo, IIRC.
Getting Soyuz ready, with its more advanced design [albeit lacking lunar-capable heat-shield] took a long time, loss of crew as it was rushed into service without sufficient testing and preparation. Helped them lose the moon race.
Meanwhile, Apollo, which would have been comprehensible to a design school of 1959, won the race. The story goes, the designer who sold Apollo to NASA had been told "if you want to get picked by NASA, just give them a big Mercury." Which is what he did. Because Mercury worked, and NASA wanted something that would fly 100%. Even if it was outdated.
The primary reason NASA won the moon race, aside from greater funding, was the more consistent political direction and support from the highest levels, as opposed to squabbling design bureaus in the USSR. NASA also created consistent operational procedures to minimize risk, whereas Soviet program was more like an experimental artillery test range. Ironically, US won because it implemented central planning better than USSR at the time.
