Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer.

The Tate conjecture asserts surjectivity of the cycle class map:

$$c^r_{\ell}(X): Z_r(X)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}\to H^{2r}_{\rm et}(X_{k^{\rm sep}},\mathbf{Q}_{\ell}(r))^{\text{Gal}(k^{\rm sep}/k)}$$

$c^r_{\ell}(X)$ factors through the group of $r$-cycles modulo algebraic equivalence, $\text{NS}^r(X)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}$. We denote by $\text{ns}_{\ell}^r(X)$ the resulting cycle map.

It would seem the Tate conjecture would then follow from surjectivity of $\text{ns}^r_{\ell}(X)$. Am I missing something up there? Chow groups of cycles modulo rational equivalence are much larger than Néron Severi groups, so this is saying Tate cycles are expected to all be images of cycles modulo algebraic equivalence. Is this correct?

It is known (Thm. of the Base) that $\text{NS}^1(X)$ is finitely generated. Is this known for arbitrary $r\ge 0$?

Is anything known about $\ker(\text{CH}^r(X)\twoheadrightarrow\text{NS}^r(X))$?