Schemes play an enormous role in all the modern theory of elliptic curves, and have done so ever since Mazur and Tate proved their theorem that no elliptic curve over $\mathbb Q$ can have a 13-torsion point defined over $\mathbb Q$.
For some additional explanation, you could look at this answer. But bear in mind that theorems on the classification of torsion, while fantastic, are just a tiny part of the theory of elliptic curves, and a tiny part of how schemes are involved. One of the most important theorems about elliptic curves is the modularity theorem, proved by Wiles, Taylor, et. al. twenty or so years ago, which implies FLT. These arguments also depend heavily on modern algebraic geometry.
Also, the proof of the Sato--Tate conjecture.
Also, all current progress on the BSD conjecture.
The underlying point is the the theory of elliptic curves is one of the central topics in modern number theory, and the methods of scheme-theoretic alg. geom. are among the central tools of modern number theory. So certainly they are applied to the theory of elliptic curves
On the other hand, you won't find it so easy to synthesize your reading on the arithmetic of elliptic curves and your reading of scheme theory. For example, even Silverman's (first) book, which is quite a bit more advanced than Silverman--Tate, doesn't use schemes. Some of the arguments can be clarified by using schemes, but it takes a bit of sophistication to see how to do this, or even where such clarification is possible or useful.
Hartshorne has a discussion of elliptic curves in Ch. IV, but it doesn't touch on the number theoretic aspects of the theory; indeed, Hartshorne's book doesn't make it at all clear how scheme-theoretic techniques are to be applied in number theory.
With my own students, one exercise I give them to get them to see how make scheme-theoretic arguments and use them to study elliptic curves is the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with good or multiplicative reduction at $p$; then prove that reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb Q}$ to the endomorphisms of the reduction of $E$ mod $p$ over $\overline{\mathbb F}$ is injective.
The proof isn't that difficult, but requires some amount of sophistication to discover, if you haven't seen this sort of thing before.
Finally:
None of the results on torsion on elliptic curves over $\mathbb Q$, or modularity, or Sato--Tate, or BSD, will be accessible to you in the time-frame of your masters (I would guess); any one of them takes an enormous amount of time and effort to learn (a strong Ph.D. student working on elliptic curves might typically learn some aspects of one of them over the entirety of their time as a student). I don't mean to be discouraging --- I just want to say that it will take time, patience, and also a good advisor, if you want to learn how schemes are applied to the theory of elliptic curves, or any other part of modern number theory.
I will try to isolate the idea of the proof of the associativity, hope this answers the unclear points. First, we are doing the following, when we use the Cayley-Bacharach link in the OP.
The definition of the sum:
We start with $O$, fixed (rational) point on some fixed cubic curve
$$
E\subset \Bbb P^2\ ,
$$
(all spaces defined over a fixed field,)
then consider two other points, $P,Q$, and use a specific receipt to define the point $P+Q$. The notations in loc. cit are rather irritating, and i will never use something like $PQ$ (for a point). The point P+Q is defined uniquely as in the "picture":
O
|
|
P+Q
|
|
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*---- P------Q
Here, points in triple joined in the picture through a line correspond to points on a line in the geometry (of the affine space where the elliptic curve also lives in). Now consider a further point $R$, we want to show:
$$(P+Q)+R = P+(Q+R)\ .$$
This means the equality of the following points "in the middle" X and X' of the diagrams:
O
|
|
P+Q --- X --------- R
| |
| |
| |
A ---- P --------- Q
and
O ----- Q+R -------- B
| |
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X' -------- R
| |
| |
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P --------- Q
(Above, starting from X, and respectively X', we have to intersect
the lines OX and OX' with the curve, the "third point" is then
$(P+Q)+R$, and respectively $P+(Q+R)$.)
So it is natural to show that both diagrams fit in the same picture:
O ----- Q+R -------- B ------- Line L3
| | |
| | |
P+Q ----- * --------- R ------- Line L2
| | |
| | |
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A ------ P --------- Q ------- Line L1
| | |
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Line Line Line
M1 M2 M3
The question is explicitly, if the points * = X and respectively * = X',
constructed as follows starting with the eight points "on the margin", $A,P,A,R,B,Q+R,O,P+Q$ do coincide:
- Consider the lines $L_1,L_2,L_3$, then on $L_1,L_3$ we already have by construction three points, let $X$ be the third point on $L_2$. Let $l_1,l_2,l_3$ be degree one homogeneous polynomials, so that the equations $l_1=0$, $l_2=0$, $l_3=0$, describe the lines $L_1,L_2,L_3$. Then the degree three polynomial $l_1l_2l_3$ defines a (degenerated) cubic.
- Consider the lines $M_1,M_2,M_3$, then on $M_1,M_3$ we already have by construction three points, let $X'$ be the third point on $M_2$.
The proof forgets now everything about $X,X'$, introduces a new point, $Y$, defined as the intersection of the lines $L_2$ and $M_2$. (A priori, this point may or may not lie on the elliptic curve. In the end, all three points $X,X',Y$ coincide.) We are now in the position now to apply in the generic case (eight distinct points) the theorem Cayley-Bacharach for the (degenerated) cubic curves
$$
\begin{aligned}
C_l &:& l_1l_2l_3&=0\ ,\\
C_m &:& m_1m_2m_3&=0\ ,
\end{aligned}
$$
and the given elliptic curve $E$.The $8+1$ points are
$A,P,A,R,B,Q+R,O,P+Q$ plus $Y$. It follows, that $Y$ is also on the cubic $E$. We get by construction $X=Y=X'$, the relation we wanted.
In case some of the points coincide, we have to use multiplicities, this leads to the solution from [Fulton, Algebraic Curves], a sort of intersection number (as part of an intersection theory) is needed.
Best Answer
There is a geometric proof of associativity in the elementary undergraduate book by Silverman and Tate Rational Points on Elliptic Curves.
The proof there is indeed along the lines you suggest of considering a pencil of cubics with nine base points, and is illustrated by a nice drawing.
The textbook derives from 1961 lectures by Tate, one of the best specialists ever in elliptic curves (he received the prestigious Abel prize in 2010).