Yes, that is true. But it is one more reason why we are not getting back to normal any time soon (in my opinion).

It will get only more difficult, in the early phases of the vaccine rollout, with patience and improved logistics, the ratio of vaccinated people improved rapidly. But what can they do next? Now comes the "stick", they will introduce more and more hurdles for the unvaccinated, but that takes time, causes resistance, and it will not increase the numbers fast enough.

All the while the variants are coming, and the existing vaccines do not show the same protection against the new variants, so there is a chance that the cycle will just continue with new variants, and new vaccines, new variants...