When reading Stacks Project I encountered an argument that shows some blow-up is affine. It seems that the situation can be generalised, but I am not very sure if I am missing something, so I would like to know if the following statement is true.
Claim:
Let $S$ be a scheme. If $\forall s\in S$, $\mathscr O_{S,s}$ is Noetherian of dimension $\le1$ (e.g. $S=\operatorname{Spec}(R)$ with $R$ Noetherian of dimension $1$) and $\mathcal I$ is a quasi-coherent sheaf of $\mathscr O_S$-ideals of finite type, then the blow-up of $S$ at $\mathcal I$ is affine (as a morphism).
Attempted proof:
Consider the blow-up $b:X\to S$ of $S$ at $\mathcal I$. By Stacks project Lemma 30.32.13, $b$ is projective, and hence proper. By stacks project Lemma 30.32.10 and stacks project Lemma 32.17.2, $b$ is affine around a neighborhood of any point $s$ of $S$ such that $\mathscr O_{S,s}$ is Noetherian of dimension $\le1$. By our assumptions on $S$, $b$ is affine. $\square$
Any thoughts or references are welcomed. Thanks in advance.