Moti Gitik proved (assuming large cardinals) that all uncountable alephs can have cofinality ω. [All uncountable cardinals can be singular, Israel J. Math. 35 (1980), 61–88] I believe this may be the model you're looking for, but I don't know what happens to non-wellorderable sets in that model. According to the abstract copied below, this model is very close to what you have in mind.
Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, we prove the consistency with ZF of the statement that every infinite set is a countable union of sets of smaller cardinality. Some other statements related to this one are investigated too.
Every wellorderable set $S$ in this model belongs to the closure of $\{\{s\}:s \in S\}$ under iterated countable unions. The proof is by induction on the cardinality of the infinite set $S$. As in the abstract, we may write $S = \bigcup_{i<\omega} S_i$ where $|S_i| < |S|$ for each $i$. By the induction hypothesis, each $S_i$ belongs to the closure of $\{\{s\}:s \in S\}$ under iterated countable unions, and thus $S$ belongs to this closure too.
Actually, every set $S$ in this model belongs to the closure of $\{\{s\} : s \in S\}$ under iterated countable unions; this is essentially what Gitik's Theorem 6.3 says.