There are three fundamental facts in play here. Probably one could also use some kind of algorithm from commutative algebra to constructively obtain an explicit algebraic relation between $f$ and $g$ given $R_1,R_2,R_3,R_4$ but we don't need to actually compute the relationship, so I'll ignore that.
Lemma I. The differential equation $\dot{\wp}^2=4\wp^3-g_2\wp-g_3$ is satisfied.
Since $\wp$ has a pole of order $2$ at zero, $\dot\wp$ has a pole of order $3$, and so we should be able to say $\dot\wp^2\sim C\wp^3$ for some constant $C$. We may calculate the first few terms of $\wp$'s Laurent expansion (and hence that of $\dot\wp$'s as well) by expanding the individual terms in $\wp$'s defining sum via geometric series formula, then interchanging order of summation. With the Laurent expansion we can determine that $C=4$, and then look at the expansion of $\dot\wp^2-4\wp^3$: we see this difference is equal to $A\wp +B+O(z)$ around $z=0$ for an appropriate choice of constants $A=-60G_4$ and $B=-140G_6$. Thus the difference between the two sides of the differential equation is a bounded entire function, which must be constant, which must be $0$ since near $z=0$ it is $O(z)$.
This is repeated in Theorem 2 here.
Lemma II. The function field of doubly periodic functions is $\Bbb C(\wp,\dot\wp)$.
Every function $f$ can be split as $f=f_{\rm even}+f_{\rm odd}$ into even and odd parts. If $f$ is an odd elliptic function with given period lattice, then $f(z)/\dot\wp(z)$ is an even elliptic function with the same period lattice since we know $\dot\wp$ is odd. Thus $f(z)=A(z)+B(z)\dot\wp$ for even elliptic functions $A$ and $B$. We want to see that even elliptic functions are rational functions in $\wp$, which means ratios of linear factors of the form $\wp(z)-w$. As $\wp$ is surjective, we can think of these factors as $\wp(z)-\wp(u)$ for poles or roots $u$ of the function $f$. This is the subject of this MSE question.
Lemma III. Transcendence degrees of field extensions are well-defined.
Let $L/K$ be any extension of fields. By Zorn's lemma there exists a subset $S\subset L$ which is algebraically independent over $K$ and for which $L/K(S)$ is algebraic, and any algebraically independent subset is contained in a maximal such one $S$. The cardinality $|S|$ is an invariant of the extension $L/K$; it does not depend on which $S$ was chosen. It is known as the transcendence degree. It is analogous to the invariant of a dimension for vector spaces (the size of a basis is independent of which basis is chosen).
See Theorem 115 in Pete L Clark's notes on Field Theory.
Assume $\wp$ is transcendental (satisfies no algebraic relation) over $\Bbb C$. We know $\Bbb C(\wp,\dot\wp)$ is an algebraic extension of $\Bbb C(\wp)$ since it's made by adjoining a square root of $4\wp^3-g_2\wp-g_3$, and so $\{\wp\}$ is a transcendence basis for $\Bbb C(\wp,\dot\wp)/\Bbb C$. This implies the extension has transcendence degree equal to one. If $f,g\in\Bbb C(\wp,\dot\wp)$ are elliptic functions, $\{f,g\}$ cannot be algebraically independent as that would imply transcendence degree $\ge2$, a contradiction.
If $\wp$ were algebraic over $\Bbb C$ then $\Bbb C(\wp,\dot\wp)/\Bbb C$ would be algebraic and so no two elements would be algebraically independent. I assume $\wp$ is transcendental for any choice of period lattice and never algebraic, but now that I think about it I've never seen that explicitly stated anywhere.
