This does not directly address the question, but I hope it will avoid some misunderstandings.

In general, when $k>2$, the L-values (for any twist by a $p$-power order character) at $s=1$ being nonzero is not enough to conclude that the p-adic L-function does not vanish identically. For example, in the case of a $p$-ordinary cusp form $f$ of even weight $k>2$, one could not garantee that the p-adic L-function of $f$ does not vanish identically until one had precisely Rohrlich's results or equivalent (since the non-central critical values are nonzero "for free", as argued by Rob in his first comment).

In fact, in that case the $p$-adic L-function is given by a power series, in $T$ say, and one does not know that it is nonzero until one can show that is does not interpolate zero infinitely many times (i.e. that it does not have infinitely many distinct zeros), and for that it clearly does not suffice to say that just one of the values of interpolation is nonzero. In order words, and I think this is the point that is the point of confusion, the formula recovering $L(f,1)$ from the value of the $p$-adic $L$-function at $T=0$ is part of an interpolation problem that we do not know a priori whether it has or not a (nonzero) solution, and hence one can not reach the conclusion as in the statement of this question.

This does not directly address the question, but I hope it will avoid some misunderstandings.

In general, when $k>2$, the L-values (for any twist by a $p$-power order character) at $s=1$ being nonzero is not enough to conclude that the p-adic L-function does not vanish identically. For example, in the case of a $p$-ordinary cusp form $f$ of even weight $k>2$, one could not garantee that the p-adic L-function of $f$ does not vanish identically until one had precisely Rohrlich's results or equivalent (since the non-central critical values are nonzero "for free", as argued by Rob in his first comment).

In fact, in that case the $p$-adic L-function is given by a power series, in $T$ say, and one does not know that it is nonzero until one can show that is does not interpolate zero infinitely many times (i.e. that it does not have infinitely many distinct zeros), and for that it clearly does not suffice to say that just one of the values of interpolation is nonzero. In order words, and I think this is the confusing point, the formula recovering $L(f,1)$ from the value of the $p$-adic $L$-function at $T=0$ is part of an interpolation problem that we do not know a priori whether it has or not a (nonzero) solution $-$ of course, now we know that it does, but not from the mere fact that $L(f,1)\neq 0$ in those cases $-$, and hence one can not reach the conclusion as in the statement of this question.

This does not directly address the question, but I hope it will avoid some misunderstandings.

In general, when $k>2$, the L-values (for any twist by a $p$-power order character) at $s=1$ being nonzero is not enough to conclude that the p-adic L-function does not vanish identically. For example, in the case of a $p$-ordinary cusp form $f$ of even weight $k>2$, one could not garantee that the p-adic L-function of $f$ does not vanish identically until one had precisely Rohrlich's results or equivalent (since the non-central critical values are nonzero "for free", as argued by Rob in his first comment).

In fact, in that case the $p$-adic L-function is given by a power series, in $T$ say, and one does not know that it is nonzero until one can show that is does not interpolate zero infinitely many times (i.e. that it does not have infinitely many distinct zeros), and for that it clearly does not suffice to say that just one of the values of interpolation is nonzero. In order words, and I think this is the point that is the point of confusion, the formula recovering $L(f,1)$ from the value of the $p$-adic $L$-function at $T=0$ is part of an interpolation problem that we de not know a priori whether it has or not a (nonzero) solution, and hence one can not reach the conclusion as in the statement of this question.

This does not directly address the question, but I hope it will avoid some misunderstandings.

In general, when $k>2$, the L-values (for any twist by a $p$-power order character) at $s=1$ being nonzero is not enough to conclude that the p-adic L-function does not vanish identically. For example, in the case of a $p$-ordinary cusp form $f$ of even weight $k>2$, one could not garantee that the p-adic L-function of $f$ does not vanish identically until one had precisely Rohrlich's results or equivalent (since the non-central critical values are nonzero "for free", as argued by Rob in his first comment).

In fact, in that case the $p$-adic L-function is given by a power series, in $T$ say, and one does not know that it is nonzero until one can show that is does not interpolate zero infinitely many times (i.e. that it does not have infinitely many distinct zeros), and for that it clearly does not suffice to say that just one of the values of interpolation is nonzero. In order words, and I think this is the point that is the point of confusion, the formula recovering $L(f,1)$ from the value of the $p$-adic $L$-function at $T=0$ is part of an interpolation problem that we do not know a priori whether it has or not a (nonzero) solution, and hence one can not reach the conclusion as in the statement of this question.