Most had Pfizer or Moderna (the two mRNA-based vaccines), but a few had Johnson & Johnson or AstraZeneca. The first two vaccines are generally preferred, because their efficacy is higher. And AstraZeneca had some controversy associated with its rollout: https://www.npr.org/sections/goatsandsoda/2021/06/01/1002067808/astrozenecas-rocky-rollout-the-woes-of-the-vaccine-of-the-world

As to the emergency use authorization, it is just a practical thing to do, in my opinion. The normal pace for vaccine rollouts is based on normal risks. When there is a much higher than normal risk, it only makes sense that approval can be speeded up.

Despite that, the testing trials for the covid vaccines were quite extensive prior to release to the general public and the rollout has been very smooth, overall, as far as I can tell. If someone is a relative hermit, I can see them waiting until the vaccine gets full FDA approval. But if they interact much with other people, the relative risk ratio seems strongly tilted towards getting vaccinated (originally this might have been true only for older people, but with the Delta variant it looks true for everyone that is eligible for the shot).