Deriving Chinese Remainder Theorem from gcd Bezout identity

Correct, if $x$ is a root of the congruences then $\, x= j\,m_1 + r_1 = -k\, m_2 + r_2\,$ has roots $\,j,k\in\Bbb Z$.

This argument reverses:$ $ if $\,j,k\,$ are integers with $\ j\, \color{#c00}{m_1}+ k\, \color{#0a0}{m_2} =\, \color{#0a0}{r_2} - \color{#c00}{r_1}\, $ then by $\rm\color{#c00}{re}\color{#0a0}{arranging}$ $\ x :=\ \color{#c00}{r_1} +\, j\ \color{#c00}{m_1}^{\phantom{|}}\ =\,\ \ \color{#0a0}{r_2}\: -\,\ k\ \color{#0a0}{m_2}\ $ is one solution of the given system of congruences because $\,x\equiv \color{#c00}{r_1}\!\!\pmod{\!\!\color{#c00}{m_1}}^{\phantom{|^|}}\!\!\!,$ $x\equiv \color{#0a0}{r_2}\!\pmod{\!\!\color{#0a0}{m_2}}.\,$ Since you've already discovered one such solution for $\,j,k,\,$ viz. $\,j=hu,\,k^{\phantom{|^|}}\!\! = hv,\,$ you need only substitute into the above rearranged CRT solution for $\,x.$

Remark $ $ Combining both directions above and adding a final gcd equivalence yields the following CRT Solvability Criterion: (which immediately generalizes to ideals)

Theorem $\ \ \left.\exists\, x\in\Bbb Z\!: \begin{align}x\equiv r_1\!\!\!\pmod{\!m_1}\\ x\equiv r_2\!\!\!\pmod{\!m_2}\end{align}\right\} \begin{array}{l}\!\iff \exists\,j,k\in\Bbb Z\!:\ j\,m_1\! + k\, m_2 =\, r_2\!-r_1 \\ \!\iff\, \gcd(m_1,\,m_2)\mid r_2 -r_1\end{array}$

Proof $ $ Clearly $\,d := \gcd(m_1,m_2)\mid r_2-r_1 \,$ is a necessary condition for the equation to have roots $\,j,k\in \Bbb Z,\,$ by $\,d\mid m_1,m_2\Rightarrow\, d^{\phantom{|}}_{\phantom{i}}\!\mid j m_1\! + km_2 = r_2 - r_1.\,$ Further this condition is also sufficient by Bezout (or, constructively, by the extended Euclidean algorithm), i.e. we can scale the Bezout equation $\, a m_1\! + b m_2 = d\,$ by $\, c = \large \frac{r_2\,-\,r_1^{\phantom{.}}}{d}\,$ to get $\,ca\,m_1\!+cb\,m_2 = r_2-r_1 \,$ so, as above, rearranging this yields a congruence system solution: $\ x\, :=\, r_1 + ca\,m_1 = r_2 - cb\,m_2$.

Thus the congruence system is solvable $\iff d=\gcd(m_1,m_2)\mid r_2-r_1, \,$ i.e. iff the pair of congruences is consistent mod their moduli gcd, and when true we can constructively read off a solution from the Bezout equation for the moduli by translating it into the equivalent system language as above, i.e. scale the Bezout equation to obtain the residue difference $\,r_1-r_2\,$ then rearrange it as above to obtain $\,x.\,$ Here is a worked example from this viewpoint, and here is some motivational examples. Summarizing we've the following simple Bezout-based CRT method for solving congruence systems

$\! \small \textbf{ scale the Bezout equation for the moduli gcd}\!$ $\small \textbf{ to get the residue difference, then }\rm\color{#c00}{re}\color{#0a0}{arrange}$

This generalizes: a system of $n$ congruences is solvable $\iff$ each pair of congruences is solvable as above, and we can solve the system by successively replacing a pair of congruences by a single equivalent congruence. By induction we eventually obtain a single congruence equivalent to the entire system.

Notice the above-linked ideal version can be expressed succinctly in terms of cosets, viz.

$$ \bbox[9px,border:1px solid #c00]{r_1\! +\! m_1\Bbb Z\,\cap\, r_2\! +\! m_2\Bbb Z \neq \phi \iff r_1-r_2 \in m_1\Bbb Z+m_2\Bbb Z}\qquad\qquad $$