$\newcommand{\E}{\mathbf{E}} \newcommand{\co}{\mathcal{O}} \newcommand{\free}{\mathrm{Free}} \newcommand{\H}{\mathrm{H}}$Here's one way of seeing the Bott-Samelson theorem. The James splitting gives an equivalence
$$\Sigma \Omega \Sigma X \simeq \bigvee_{n>0} \Sigma X^{\wedge n},$$
so you find that if $k$ is a field, then the reduced homology $\H_\ast(\Omega \Sigma X;k)$ (which is the homology of the suspension spectrum of $\Omega \Sigma X$) is isomorphic to $\bigoplus_{n\geq 0} \H_\ast(X^{\wedge n}; k) \cong \bigoplus_{n\geq 0} \H_\ast(X; k)^{\otimes n}$. This is the tensor algebra on $\H_\ast(X; k)$; one observes that this isomorphism is actually multiplicative, too (it comes from the James splitting for $\Sigma^\infty_+ \Omega \Sigma X$ being multiplicative), so $\H_\ast(\Omega \Sigma X;k)$ is multiplicatively isomorphic to the tensor algebra on $\H_\ast(X; k)$.

I'll write $\E_k$ for what you call $D_k$, so that the space $\Omega \Sigma X$ is the free $\E_1$-space on $X$. In general, if $\co$ is an operad in spaces, then the $E$-homology $E_\ast(\free_\co(X))$ of the free $\co$-space on $X$ is given by the homotopy of $E \wedge \free_\co(X) \simeq \free_\co(E\wedge X)$, where the right hand side is the free $\co$-algebra on $E\wedge X$ in $E$-modules. This final equivalence is eseentially a consequence of the fact that the free functor is defined as a homotopy colimit, and these commute with smash products. The homotopy of this spectrum will be (in nice cases, e.g., homology with field coefficients) the free $E_\ast$-module generated by $\co$-$E$-Dyer-Lashof operations on $E_\ast X$.

One way to understand these Dyer-Lashof operations is as follows. There is a *stable* Snaith splitting which works more generally for any operad in spaces as above (that specializes to the stable version of the James splitting for $\co = \E_1$):
$$\Sigma^\infty_+ \free_\co(X) \simeq \bigvee_{n\geq 0} (\co(n)_+ \wedge \Sigma^\infty X^{\wedge n})_{h\Sigma_n},$$
where $\co(n)$ are the spaces in $\co$. One can prove this essentially formally. The $\co$-$E$-Dyer-Lashof operations are encoded in the $E$-homology of $\co(n)$ --- this is the non-formal component of computing the $E$-homology of free $\co$-spaces. If $\co = \E_k$, then $\co(n) = \mathrm{Conf}_n(\mathbf{R}^k)$; when $k=1$, you find that there are no interesting Dyer-Lashof operations (other than "take powers"), so you get the Bott-Samelson result from these general considerations. For higher $k$, these Dyer-Lashof operations get a lot more interesting.

The homology of $C_{n+1}$ spaces. But this is only for $n > 0$ (so it doesn't cover $C_1 = D_1$ in your notation). And skimming the proof, it seems to take as granted that $H_*(\Omega \Sigma X)$ is the free associative algebra on $\tilde{H}_*(X)$ (he calls them $W_0$-algebras). But I guess my point is that it's not automatic that $H_*(P(X)) = H_*(P)(H_*(X))$ in general, you have to work for it – Cohen's proof is an example of that: the homology of the free $D_n$-algebra is not the free $H_*(D_n)$-algebra over $\mathbb{F}_p$. $\endgroup$ – Najib Idrissi Aug 22 '19 at 16:47