Excision exact sequence for divisor class group on locally Noetherian scheme with no extra assumptions

Question

Let $X$ be an arbitrary locally Noetherian scheme, and let $Z \subset X$ be an arbitrary closed subset. Let $U = X - Z$. Denote by $i: Z \to X$ and $j: U \to X$ the inclusions. Since $X$ is locally Noetherian, the closed subset $Z$ admits a decomposition as a locally finite union of irreducible components, $Z = \cup_{i \in I} Z_i$. I claim that the sequence of (Weil) divisor class groups $$\mathbf{Z}^I \overset{i_*}{\to} \mathrm{Cl}(X) \overset{j^*}{\to} \mathrm{Cl}(U) \to 0$$ is exact. Here the map $i_*$ is defined to take the $(i \in I)$-th basis element of $\mathbf{Z}^I$ to the class $[Z_i]$ if $Z_i$ is of codimension $1$ in $X$, and to zero otherwise. The surjectivity of $j^*$ is easy and is proved in this level of generality in EGA IV part 4 section 21. However, this section of EGA IV (which is the very last one and is the one that deals with Cartier and Weil divisors) surprisingly does not address exactness in the middle term. Perhaps this is because Grothendieck decided to leave this question for SGA, so I checked SGA 6. I don't know anything about the $K$ groups that are dealt with in there, but anyway this exact sequence is stated in SGA 6 for Chow groups of varieties. Similarly, other references on intersection theory (including Fulton) only deal with schemes over a field. I checked the Stacks project, and the closest thing I could find is https://stacks.math.columbia.edu/tag/02RX, which specializes to the codimension 1 case to give the exactness I want under some very mild catenary assumptions. I understand at least one reason why the catenary assumption is needed for that lemma (so that rational equivalence can even make sense for Chow groups of cycles of arbitrary dimension). It seems to me like when you are dealing with codimension 1 cycles, this is no longer necessary (which is why there are no such catenary assumptions in the basic theory of divisors, as far as I understand it), and the exact sequence above is true without any assumptions except for "locally Noetherian" on $X$: assuming that $X$ is integral for convenience, if you have a Weil divisor on $X$ whose class is trivial upon restriction to $U$, you can just subtract the divisor of the rational function on $U$ that trivializes that divisor, and end up with a divisor that is supported outside $U$, i.e. on $Z$. I am just worried that I have fooled myself, because I cannot find this very basic lemma stated in this level of generality anywhere in the literature. Is this correct as stated, or does someone know a counterexample / reason my proof is flawed ?