[I will also answer my question since I read Kirwan's book on complex algebraic curves and I thought it was quite enlightening (regarding the complex case).]
I will state a few things about complex elliptic curves and then discuss the case of an arbitrary field using arguments from logic.
Complex elliptic curves "are" torus
It's not too difficult to see that the Weierstrass ℘ function is a meromorphic function on the torus $\mathbb{C}/\Lambda$ (seen as a compact Riemann surface).
Studying a bit compact Riemann surfaces, we learn that meromorphic functions on them are quite restrained. More specifically in this case, combining $℘$ and $℘'$ in a way that eliminates the poles yields a holomorphic function on the torus, which must be constant as per compacity and maximum modulus principle (or Liouville Theorem).
Thus, we can show that we have an isomorphism of Riemann surfaces between a torus and a complex elliptic curve (both being parametrized by $\Lambda$).
This correspondence is also a group isomorphism
We have a Riemann surface isomorphism $z \mapsto (℘(z), ℘'(z))$ (extended by mapping the pole to the point at infinity $\infty$ of the curve).
The expression of that isomorphism shows that integrating the differential form $\frac{dx}{y}$ along a path from $\infty$ to a point $P$ we can recover $z$, giving an inverse map $u$ from the curve to the torus.
If $D$ is a line in $P_2(\mathbb{C})$, it intersects a curve in 3 points $A, B, C$ and we'd like to show $f(D) := u(A) + u(B) + u(C) = 0 \mod \Lambda$ (*).
The "trick" here is to first show that $f(D) = 0$ for all lines $D$ of the form $by+cz=0$.
Those lines $D_{[b:c]}$ are parametrized by $P_1(\mathbb{C})$ in such a way that we get a holomorphic map $[b:c] \mapsto f(D_{[b:c]})$.
By Liouville theorem, this map must be constant, hence equal to its value on $[0:1]$ which is $0 \mod \Lambda$ as it corresponds to the case $A = B = C = \infty$.
We get $f(D) = 0$ for any line $D$ by considering lines of the form $\alpha x + \beta (by+cz)$ and reiterating the same kind of construction varying $[\alpha:\beta] \in P_1(\mathbb{C})$.
This shows the correspondance between the torus and the elliptic curve is a group morphism.
Subgroup of points of order $n$
The result stated in the question is obvious in the case of complex elliptic curves, given the group isomorphism with a torus.
Elliptic curves over an algebraically closed field $K$
In this case, we can't refer to Riemann surfaces and we don't have such thing as a torus.
However, the result stated about the subgroup of points of order $n$ is an algebraic result.
In particular, it can be stated in $ACF_0$ (the first order theory of algebraically closed fields of characteristic $0$) (*)
As this theory is complete, this result is true over any such field of characteristic $0$.
By compacity argument (of first order logic), this result is still true in characteristic $p > 0$ except for potentially finitely many characteristics.
The result expressed in the question make the restriction on $p$ more precise, which is possible by studying from scratch the problem in an algebraic way.
(*) Actually, for a fixed $n_0 \in \mathbb{N}$, the sentence The group of points of order $n$ has cardinal $n^2$ for all $n \leq n_0$ is expressible in the language of fields and from that we get the structure of those subgroups.