Completely false. There's several groups that might want the vaccine that can't get it. Including: children and the immuno compromised.
32 kids were hospitalized in one week in Alabama last month. A state with half the population of NYC.
https://www.al.com/coronavirus/2021/07/infant-among-children...
Are you referring to the mRNA vaccine? If so, source for this? I see lots of articles that question whether the vaccines are as effective for immuno compromised people, but none that say they shouldn't take the mRNA vaccine. In fact, I see many studies testing the efficacy of mRNA vaccines for immuno-compromised people, which would seem to indicate we think it's safe. I only know one immuno-compromised person, and she got the vaccine.
Like I said, there's practically no data on the mRNA side, particularly with how "immunocompromised" is a very broad term meaning many different underlying issues. Most of the data I've seen is focused on the efficacy rates, rather than safety from complications. IMO it's fair that immunocompromised individuals want to wait a bit when it comes to an entirely new form of vaccine.
The non mRNA vaccines obviously have potential issues for the kinds of people that could die from even a normal flu.