The key to those harmonies is they were recorded old-school, with everybody standing around an omnidirectional microphone. You get a natural blend that way that may be impossible to match when each part is sung separately, due to the fact that the all-at-once mix is entirely acoustical.
It's common practice nowadays to use both a short and a long reverb, with different send ratios for lead versus background vocals. The lead may have just as much reverb as the BGVs, but it'll typically have more of the short and less of the long.