The following fact is extremely well known:

Fact.Let $Y$ be a geometrically irreducible variety (not necessarily smooth or proper) over a finite field $k$. Then there is a constant $B$, depending only on cohomological invariants of the base change of $Y$ to an algebraic closure of $k$, such that: if $l$ is any finite extension of $k$ satisfying $|l|>B$, then $Y(l)$ is non-empty.

Unfortunately, I can't find any simple reference for this well-known fact. If $Y$ is projective (and possibly singular), then it follows from the Lang-Weil estimates. If $Y$ is smooth and proper over $k$, then it follows from Deligne's proof of the Weil conjectures. Given that we don't mind replacing $Y$ with an open subvariety, there's a plausible argument for reducing the general case to one of these cases.

Also, the phrase "depending only on cohomological invariants of..." is rather vague, especially when $Y$ might be non-proper and singular. So maybe it's better to stick to more concrete statements, such as that $B$ should remain constant in certain types of families.

I would like to cite this fact without having to devote half a page to deducing it from something else. So my question is simply:

What is a good reference for the fact stated above?