I'm sure there are easier and better ways to think about this, but here's how I like to think about it.

Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $\pi: \mathrm{Sch}_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$. Also, let's consider (hypercomplete) sheaves of anima. Then $\pi^\ast$ has a left adjoint $\pi_\natural$, and (for our given scheme $X$) $\pi_\natural(X)$ is a condensed anima that "is" the etale homotopy type of $X$. Concretely, if $X_\bullet\to X$ is a proetale hypercover by w-contractible $X_\bullet$, then $\pi_\natural(X)$ is represented by the simplicial extremally disconnected profinite set $\pi_0(X_\bullet)$. By adjunction, there is a natural map $X\to \pi^\ast \pi_\natural (X)$.

Now giving a $\mathbb Q_\ell$-local system $\mathbb L$ (same for other coefficient rings) is the same as giving a map $X\to \pi^\ast B\mathrm{GL}_n(\mathbb Q_\ell)$, i.e. equivalently a map $\pi_\natural(X)\to B\mathrm{GL}_n(\mathbb Q_\ell)$, i.e. a $\mathbb Q_\ell$-local system on the condensed anima $\pi_\natural(X)$.

Now what is homology of $X$ with coefficients in $\mathbb L$? One definition uses the formalism of solid $\mathbb Q_\ell$-sheaves $D_\blacksquare(X,\mathbb Q_\ell)$ (no reference for them in the case of schemes yet, sorry!). In that setting, pullback $D_\blacksquare(\ast,\mathbb Q_\ell)\to D_\blacksquare(X,\mathbb Q_\ell)$ has a left adjoint, which takes $\mathbb L$ to the homology of $X$ with coefficients in $\mathbb L$. Here $D_\blacksquare(\ast,\mathbb Q_\ell)$ is the "usual" derived category of solid $\mathbb Q_\ell$-modules.

Working with the condensed anima $\pi_\natural(X)$ instead, one can apply a similar procedure, and the two answers will agree, so indeed the homology of $X$ with coefficients in $\mathbb L$ agrees with the homology of $\pi_\natural(X)$ with coefficients in $\mathbb L$.

In practice, if $X$ is sufficiently nice, then these homology groups will be finite-dimensional over $\mathbb Q_\ell$ and dual to cohomology, so this would indeed follow from the usual statements about cohomology. For general $X$, this statement about homology is however slightly finer (as cohomology is always the dual of homology, but not conversely).

I'm sure there are easier and better ways to think about this, but here's how I like to think about it.

Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $\pi: \mathrm{Sch}_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$. Also, let's consider (hypercomplete) sheaves of anima. Then $\pi^\ast$ has a left adjoint $\pi_\natural$, and (for our given scheme $X$) $\pi_\natural(X)$ is a condensed anima that "is" the etale homotopy type of $X$. Concretely, if $X_\bullet\to X$ is a proetale hypercover by w-contractible $X_\bullet$, then $\pi_\natural(X)$ is represented by the simplicial extremally disconnected profinite set $\pi_0(X_\bullet)$. By adjunction, there is a natural map $X\to \pi^\ast \pi_\natural (X)$.

Now giving a $\mathbb Q_\ell$-local system $\mathbb L$ (same for other coefficient rings) is the same as giving a map $X\to \pi^\ast B\mathrm{GL}_n(\mathbb Q_\ell)$, i.e. equivalently a map $\pi_\natural(X)\to B\mathrm{GL}_n(\mathbb Q_\ell)$, i.e. a $\mathbb Q_\ell$-local system on the condensed anima $\pi_\natural(X)$.

Now what is homology of $X$ with coefficients in $\mathbb L$? One definition uses the formalism of solid $\mathbb Q_\ell$-sheaves $D_\blacksquare(X,\mathbb Q_\ell)$ (no reference for them in the case of schemes yet, sorry!). In that setting, pullback $D_\blacksquare(\ast,\mathbb Q_\ell)\to D_\blacksquare(X,\mathbb Q_\ell)$ has a left adjoint, which takes $\mathbb L$ to the homology of $X$ with coefficients in $\mathbb L$. Here $D_\blacksquare(\ast,\mathbb Q_\ell)$ is the "usual" derived category of solid $\mathbb Q_\ell$-modules.

Working with the condensed anima $\pi_\natural(X)$ instead, one can apply a similar procedure, and the two answers will agree, so indeed the homology of $X$ with coefficients in $\mathbb L$ agrees with the homology of $\pi_\natural(X)$ with coefficients in $\mathbb L$.

In practice, if $X$ is sufficiently nice, then these homology groups will be finite-dimensional over $\mathbb Q_\ell$ and dual to cohomology, so this would indeed follow from the usual statements about cohomology. For general $X$, this statement about homology is however slightly finer (as cohomology is always the dual of homology, but not conversely).

Edit: Here is a way to phrase the answer so that it does not involve a reference to $D_\blacksquare(X,\mathbb Q_\ell)$. One can instead use the full derived category of pro-etale $\mathbb Q_\ell$-sheaves $D(X_{\mathrm{proet}},\mathbb Q_\ell)$; then, just like for sheaves of anima, the pullback along $f: X_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$ has a left adjoint $f_\natural$, and $f_\natural \mathbb L\in D(\ast_{\mathrm{proet}},\mathbb Q_\ell)$ is a complex of condensed $\mathbb Q_\ell$-vector spaces that can be considered as the homology of $\mathbb L$. Again, a similar construction can be done for $\pi_\natural(X)$, and these two notions of homology agree.

On the other hand, I expect that it is extremely difficult to compute this notion of homology even for $X=\mathbb P^1_k$ for $k$ an algebraically closed field. However, passing to the solidification, one can compute it in practice, and I guess it usually agrees with the homology of the Artin--Mazur pro-(homotopy type).