Though it is perhaps not an "answer" as such, let me try to explain some intuition.
In certain settings, it is possible to formulate subtle analogues of Mazur's conjecture
for nonvanishing of central values (think Cornut-Vatsal) from the refined conjecture
of Birch and Swinnerton-Dyer via Iwasawa theory. A general method is explained in
section 4 of Coates-Fukaya-Kato-Sujatha, " Root numbers, Selmer groups, and
non-commutative Iwasawa theory" (available at http://www.math.tifr.res.in/~sujatha/root.pdf).
CFKS consider the setting of the so-called False-Tate curve extension, but
a similar (and simpler) set of arguments can be used to deduce an analogous
conjecture for the setting of the ${\bf{Z}}_p^2$ of an
imaginary quadratic field. To be slightly more precise, fix a rational prime $p$.
Fix an eigenform $f \in S_2(\Gamma_0(N))$ with $(N,p)=1$. Fix an imaginary
quadratic field $k$ with discriminant prime to $pN$. Let $k_{\infty}$ denote the
${\bf{Z}}_p^2$-extension of $k$, which is the compositum of the cyclotomic
${\bf{Z}}_p$-extension $k^c$ with the anticyclotomic ${\bf{Z}}_p$-extension $k^a$.
Write $\lambda_f(k)$ to denote the cyclotomic $\lambda$-invariant associated to
$f$, with $\mu_f(k)$ the cyclotomic $\mu$-invariant. Let $\mathcal{W}$ be any finite
order character of $\operatorname{Gal}(k_{\infty}/k)$. Such a character can always be written as
a product of characters $\rho \cdot \chi$, where $\rho$ is a finite order character
of $\operatorname{Gal}(k^a/k)$, and $\psi$ is a finite order character of $\operatorname{Gal}(k^c/k)$. What the
CFKS conjecture predicts, very roughly, is the following assertion. Assume that
$\mu_f(k)=0$, and fix a finite order character $\rho$ of $\operatorname{Gal}(k^a/k)$. Let $\Psi$
denote the set of finite order character of $\operatorname{Gal}(k^c/k)$. Then, assuming the
refined Birch and Swinnerton-Dyer conjecture, \begin{align*}
\sum_{\psi \in \Psi} \operatorname{ord}_{s =1}L(f \times \rho \cdot \psi, s) &\leq \lambda_f(k).
\end{align*} So, what does this tell us? Well for one, it tells us that Rohrlich
nonvanishing (at least in this setting) should be a general phenomenon.
One can make this intuition slightly more precise via the following heuristic
argument. View any finite extension of $k^c$ over $k$ as a totally imaginary
quadratic extension of its maximal totally real subfield. Suppose that the
nonvanishing theorem of Cornut-Vatsal were uniformly effective (in the sense
that their $n$ sufficiently large could be replaced by some absolute $n_k$ that does not
grow as we ascend the cyclotomic tower). Then, invoking their result systematically
and decomposing via Artin formalism, we should (I think) expect the following behaviour.
Let $\epsilon(f/k, s)$ denote the root number of the Rankin-Selberg $L$-function
$L(f/k, s)$. Given $\rho$ a finite order character of $\operatorname{Gal}(k^a/k)$ of conductor greater that $p^{n_k}$, we should have:

\begin{align*}
\sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 0 \text{ if $\epsilon(f/k, 1) = 1;$} \end{align*}

\begin{align*}
\sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 1 \text{ if $\epsilon(f/k, 1) = -1;$} \end{align*}

Sorry to have skipped steps, or if this is perhaps somewhat unclear in places. The main idea is that there are subtle generalizations of Mazur's conjecture to the types of settings that you will likely want to consider. These generalizations suggest that one should expect generic nonvanishing à la Rohrlich, even in the case where the root number at the bottom is $-1$. And though it is not a priori clear, it might be possible to use these sorts of ideas to obtain the general formulation of Goldfeld's conjecture that you ask for.

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