A smooth point has the exact finite-type excision pair

Choose an affine open in the smooth neighbourhood containing $x$ and shrink within its relative-dimension stratum. It is smooth separated finite type over $k$, pure of some dimension $e$. In the usual finite-type setting, purity of $X$ and [F2] give $e=d$. The same conclusion holds if only qcqs $X$ is assumed: let $T$ be the reduced irreducible component through the chosen local smooth component, with function field $K$. Its finite-type nonempty open has dimension $e={\mathrm{trdeg}}_{k}K$. Every affine coordinate domain on $T$ is a subalgebra of $K$; any finite strict prime chain in it remains strict in some finitely generated $k$-subalgebra, by choosing one witness for each strict inclusion. That subalgebra has dimension at most ${\mathrm{trdeg}}_{k}K=e$. Every finite specialization chain in $T$ lies in an affine open containing its most special point. Hence $\dim T\le e$, while its open of dimension $e$ gives the reverse inequality. Purity of $X$ gives $\dim T=d$, so $e=d$. Denote the resulting neighbourhood by $V$.

Since $x$ is closed in $V$, [F2] gives finite $L\mathrm{/}k$; perfectness makes it separable. Thus both $V$ and $\mathrm{Spec}L$ belong to the smooth separated finite-type category, and the dimension formula gives codimension $d$. The open $V\setminus \{x\}$ is quasi-compact because $V$ is Noetherian. To prove $X\setminus \{x\}$ quasi-compact without imposing global Noetherianness, take any open cover of it. Together with $V$ it covers quasi-compact $X$, so finitely many members plus $V$ suffice. Finitely many members also cover quasi-compact $V\setminus \{x\}$. These two finite selections cover $X\setminus \{x\}$. Quasi-separatedness is inherited by open subschemes.

The two opens $V$ and $X\setminus \{x\}$ cover $X$, with intersection $V\setminus \{x\}$. They form a Nisnevich two-open cover, and all four objects are qcqs by step 2.1. Apply the actual derived Nisnevich descent of [F1] and then its conditional excision theorem. This supplies exactly the claimed restriction-induced support equivalence and support-to-global compatibility for every nonnegative weight.