Motivation: When I was young(er), I was once shocked to learn that, for $X$ a scheme of finite type, $\Gamma(X,\mathscr{O}_X)$ can fail to be of finite type. Now that I am no longer so young and bashful, I followed von Neumann's advice and “got used to it”. But what if, instead of getting used to it, we tried to generalize the “finite type” property so that the expected facts work?
Question: Let us define a property $\mathbf{F}$ of schemes over a field $k$, generated by the following conditions:
affine space $\mathbb{A}^n_k$ satisfies $\mathbf{F}$,
if $X,Y$ satisfy $\mathbf{F}$ then $X\times_{\operatorname{Spec}k} Y$ satisfies $\mathbf{F}$,
if $X$ satisfies $\mathbf{F}$, then any closed subscheme $Z \subseteq X$ satisfies $\mathbf{F}$,
if $X$ satisfies $\mathbf{F}$, then any open subscheme $U \subseteq X$ satisfies $\mathbf{F}$,
if $X$ is a $k$-scheme and $U_1,\ldots,U_n$ are finitely many open subschemes of $X$ which cover $X$ and satisfy $\mathbf{F}$, then $X$ satisfies $\mathbf{F}$,
if $X$ satisfies $\mathbf{F}$, then $\operatorname{Spec}\Gamma(X,\mathscr{O}_X)$ satisfies $\mathbf{F}$,
can we characterize the $k$-schemes which satisfy $\mathbf{F}$?
(Feel free to answer the question, instead, for the property $\mathbf{F}'$ adding any “reasonable” stability conditions which are satisfied by schemes of finite type, or maybe a generalization of the last condition, if it makes the answer easier or you think it's more natural.)
At the very least, can we give a decent necessary condition for satisfying $\mathbf{F}$? (Unless I made an embarrassing mistake, being of finite type is a sufficient condition.)
NB: this question is somewhat related.
