I don't know about your argument,but I do think there is a simple symmetry based argument that avoids hacking around with densities. This just generalizes the simple observation that if X,Y i.i.d then E(X|Z = X+Y) = Z/2. When the correlation between X and Z is of form say,$sqrt{ p/(p+q)}$, then they can be represented as $Z = X_1 + ... + X_p + Y_1 + .... + Y_q$ where the X_i and Y_j are all i.i.d. Gaussian and $X= X_1 + ... + X_p$. In this case by symmetry all E(X_i|Z) and E(Y_j| Z) are the same and therefore $E(X |Z) \frac p {p+q} Z}. If necessary, you can perturb slightly to achieve that condition and do a simple limiting argument.

I don't know about your argument,but I do think there is a simple symmetry based argument that avoids hacking around with densities. This just generalizes the simple observation that if X,Y i.i.d then E(X|Z = X+Y) = Z/2. When the correlation between X and Z is of form say,$\sqrt{ p/(p+q)}$, then they can be represented as $Z = X_1 + ... + X_p + Y_1 + .... + Y_q$ where the X_i and Y_j are all i.i.d. Gaussian and $X= \frac {X_1 + ... + X_p} {\sqrt p}$. In this case by symmetry all E(X_i|Z) and E(Y_j| Z) are the same and therefore $E(X |Z) \frac p {p+q} Z$. If necessary, you can perturb slightly to achieve that condition and do a simple limiting argument.

I don't know about your argument,but I do think there is a simple symmetry based argument that avoids hacking around with densities. This just generalizes the simple observation that if X,Y i.i.d then E(X|Z = X+Y) = Z/2. When the correlation between X and Z is rational, say, p/(p+q), then they can be represented as $Z = X_1 + ... + X_p + Y_1 + .... + Y_q$ where the X_i and Y_j are all i.i.d. Gaussian and $X= X_1 + ... + X_p$. In this case by symmetry all E(X_i|Z) and E(Y_j| Z) are the same and therefore $E(X |Z) \frac p {p+q} Z}. If necessary, you can perturb slightly to achieve that condition and do a simple limiting argument.

I don't know about your argument,but I do think there is a simple symmetry based argument that avoids hacking around with densities. This just generalizes the simple observation that if X,Y i.i.d then E(X|Z = X+Y) = Z/2. When the correlation between X and Z is of form say,$sqrt{ p/(p+q)}$, then they can be represented as $Z = X_1 + ... + X_p + Y_1 + .... + Y_q$ where the X_i and Y_j are all i.i.d. Gaussian and $X= X_1 + ... + X_p$. In this case by symmetry all E(X_i|Z) and E(Y_j| Z) are the same and therefore $E(X |Z) \frac p {p+q} Z}. If necessary, you can perturb slightly to achieve that condition and do a simple limiting argument.