This is not a new idea; indeed, almost all work on the BSD conjecture has proceeded via a study of the associated modular form. In particular, one
has the work of Gross--Zagier and Kolyvagin (see e.g. here), which verifies
BSD for those $E$ such that $L(E,s)$ vanishes to order zero or one at $s = 1$,
and the work of Kato (described very briefly here), which give another proof of BSD for those $E$ such that $L(E,s)$ is non-vanishing at $1$
(and much more besides).
The difficulty with proving BSD (for definiteness, let's consider the form which says that the order of vanishing of $L(E,s)$ at $s = 1$ is equal to the
Mordell--Weil rank of $E$)
is that there is no obvious relationship between
the $L$-function and the points on $E$ at all. The modular parameterization
provided by the modularity theorem for elliptic curves does provide a more direct relationship between the two sides of the conjecture, because the modular
form associated to the elliptic curve has both a relationship to the $L$-function, via a Mellin transform, and a relationship to the elliptic curve via the modular parameterization, but it is still very difficult to extract anything related to BSD out of this relationship.
Gross and Zagier use Heegner points on the modular curve to generate an infinite order point on $E$ in the case when $L(E,s)$ vanishes
to first order at $s = 1$. Kolyvagin then makes an elaborate argument (a so-called Euler system argument) to show that in fact in this case $E$ has Mordell--Weil rank exactly one.
Gross and Zagier don't use the Mellin transform formula for the $L$-function, but a different formula, due to Rankin.
In Kato's work, he also uses a version of the Rankin--Selberg formula, and combines it with a different Euler system argument, generalizing a construction due to Beilinson.
In both cases, if you work through the arguments, you find that the relationship between the $L$-function and the group of rational points on the elliptic curve, while it is mediated by the modular parameterization, is nevertheless very subtle and indirect. Many experts have thought about how to push things further, with no success so far.