Roots of unity, almost radically closed fields, and the Artin--Schreier theorem

Motivation: I want to classify all fields $K$ which satisfy the property 1. (below). I also want to classify all fields $K$ which satisfy the property 2. (below). Alternatively, I would be happy with showing that there are interesting fields satisfying 1. or 2. (apart from real closed fields, algebraically closed fields, and subfields of the algebraic closure of finite fields): If $\alpha$ is algebraic over $K$, then for every $n \in \mathbb{N}_+$ there is some root of unity $\zeta \in K(\alpha)$ and an element $\beta \in K(\alpha)$ such that $\zeta \alpha = \beta^n$. Stronger: For every finite field extension $K \subseteq L$, the quotient of $L^\times$ by its maximal divisible subgroup is a torsion group. My Galois theory is a bit rusty. Here are my thoughts: This feels somewhat related to the Artin-Schreier theorem: If $K \subseteq \overline{K}$ is a finite field extension, where $\overline{K}$ is the algebraic closure of $K$, then either $K = \overline{K}$ or $K$ is real closed (in particular, not algebraically closed, of characteristic $0$ and $\overline{K} = K(\sqrt{-1})$). If it is possible to show the following: Assume $K \subseteq \overline{K}$ is generated by roots of unity (possibly infinitely many), then $K$ is real closed, algebraically closed, or the subfield of the algebraic closure of a finite field. Then in characteristic $0$, I think, it would follow that 1.$\iff$2 is precisely satisfied for the real closed fields and the algebraically closed fields using Problem 2(ii) in https://www.math.uni.wroc.pl/~kkrup/stable_groups_list7.pdf (which I'm also currently unable to solve): If for all finite field extensions $K \subseteq L$ and all $n \in \mathbb{N}_+$ the map(s) $X \mapsto X^n$ (and $X \mapsto X^p -X$ in characteristic $p$) are surjective maps $L^\times \to L^\times$, then $K$ is already algebraically closed. I would appreciate anything related to 1., 2., 3. or 4. See also my post on Math Overflow.