[Apologies ahead of time that this question is (1) quite heavy in laying out notations, and (2) of the can-you-spot-my-error variety.]
I am trying to recast the Eichler–Shimura isomorphism as a simple consequence of generally known log-smooth cases of mixed Hodge theory* applied to the complete modular curve. (*See [KN] Kato–Nakayama, Kodai Math. J. 1999.) In particular, the map of the isomorphism should be constructed in a way that avoids restriction to the open modular curve. But I'm getting tripped up somewhere, and I would like some help cleaning up my misconceptions.
First let me recall what the Eichler–Shimura isomorphism says.
Fix a congruence subgroup $\Gamma \subseteq SL_2(\mathbf{Z})$, small enough to avoid any stackiness issues. Let $Y = \mathfrak{h}/\Gamma$ be the open modular curve (considered as a Riemann surface), with universal elliptic curve $f : E \to Y$, compactification $X$, and cusps $S = X-Y$. On the one hand, if $\tilde E \to X$ is the Néron model of $E$ with unit section $e$, we let $\omega = e^*\Omega^1_{\tilde E/X}$, a line bundle on $X$, so that Kodaira–Spencer gives $\omega^2 \cong \Omega^1_X(S)$. We also put $\mathcal{H}_{dR} = H^1_{dR}(E/Y)$. On the other hand, we let $\mathcal{H}_B = R^1f_*\underline{\mathbf{C}}$, a local system on $Y$. The relative de Rham theorem gives $\mathcal{H}_{dR} \cong \mathcal{H}_B \otimes \mathcal{O}_Y$. Then the Eichler–Shimura isomorphism has the form
$\alpha \oplus \iota\alpha : H^0(X,\omega^{k+2}) \oplus \iota H^0(X,\omega^k \otimes \Omega^1_X) \cong H^1(Y,Sym^k\mathcal{H}_B)$.
(Here $\iota$ denotes complex conjugation, but by duality one should really think of the second summand as $H^1(X,\omega^{-k})$.) In short, the morphism $\alpha$ is defined by restricting to $Y$, followed by embedding $\omega^k \hookrightarrow Sym^k \mathcal{H}_{dR} \cong Sym^k \mathcal{H}_B \otimes \mathcal{O}_Y$ (tensored with the remaining $\omega^2|_Y \cong \Omega^1_Y$), followed by the connecting map for the short exact sequence $\mathcal{H}_B \otimes (0 \to \underline{\mathbf{C}} \to \mathcal{O}_Y \to \Omega^1_Y \to 0)$.
Now let me review the log situation.
On the differential side, equip $X$ with fs log structure coming from $S$, so that $\Omega^1_{X,\log} = \Omega^1_X(S)$. Denote $\bar f : \bar E \to X$ the log-smooth compactification of $f$ to a log elliptic curve, with unit section $\bar e$; the line bundle $\omega$ agrees with $\bar e^*\Omega^1_{\bar E/X}$. We also set $\bar{\mathcal{H}}_{dR} = H^1_{log.dR}(\bar E/X)$, equipped with its Gauss–Manin connection $\nabla : \bar{\mathcal{H}}_{dR} \to \bar{\mathcal{H}}_{dR} \otimes \Omega^1_{X,\log}$, and the Hodge–de Rham spectral sequence gives $0 \to \omega \to \bar{\mathcal{H}}_{dR} \to \omega^{-1} \to 0$.
On the Betti side, there is a canonical proper surjection $\tau : X^{\log} \to X$, with $X^{\log}$ a real manifold-with-boundary that boils down to replacing each cusp with a circle on the boundary; in particular $\tau^{-1}(Y) \stackrel\sim\to Y$ is homotopy equivalent $X^{\log}$. Since $(\bar{\mathcal{H}}_{dR},\nabla)$ is nilpotent, one can attach to it the local system $\bar{\mathcal{H}}_B = (\tau^*\bar{\mathcal{H}}_{dR})^{\nabla=0}$ on $X^{\log}$. Due to the homotopy equivalence property, I believe that restriction yields $H^*(X^{\log},Sym^k\bar{\mathcal{H}}_B) = H^*(Y,Sym^k\mathcal{H}_B)$.
Let us try to reconstruct $\alpha$ without restricting to $Y$ until the very end. First one embeds $\omega^k \hookrightarrow Sym^k\bar{\mathcal{H}}_{dR}$, tensored with $\omega^2 \cong \Omega^1_{X,\log}$. Next, the connecting homomorphism step is replaced by the first nontrivial map in the hypercohomology short exact sequence $0 \to H^0(X,Sym^k\bar{\mathcal{H}}_{dR} \otimes \Omega^1_{X,\log}) \to H^1_{dR}(X,Sym^k\bar{\mathcal{H}}_{dR}) \to H^1(X,Sym^k\bar{\mathcal{H}}_{dR}) \to 0$. Next, one has the de Rham theorem with coefficients of [NS], $H^1_{dR}(X,Sym^k\bar{\mathcal{H}}_{dR}) \cong H^1(X^{\log},Sym^k\bar{\mathcal{H}}_B)$. Finally one applies the homotopy equivalence to the right hand side.
This would appear to prove a short exact sequence
$0 \to H^0(X,Sym^k\bar{\mathcal{H}}_{dR} \otimes \omega^2) \to H^1(Y,Sym^k\mathcal{H}_B) \to H^1(X,Sym^k\bar{\mathcal{H}}_{dR}) \to 0$,
with the groups we want, $H^0(X,\omega^{k+2})$ and $H^1(X,\omega^{-k})$, appearing as sub and quotient objects of the first and last terms, respectively. However, there is definitely spurious information here: $H^1(Y,Sym^k\mathcal{H}_B)$ would apparently see not only holomorphic forms of weight $k+2$, but all nearly holomorphic ones of weight $k+2$ and type $\leq k$. This state of affairs is not compatible with the classical statement. So what is going on here?
