Since $7$ is coprime to the other bases, that constraint is entirely additional to the other constraints, and the fact that you get the smallest number from the other constraints fitting that constraint is indeed a coincidence.
The three middle constraint bases carry the potential for eliminating any possible solutions, as $6$ is not coprime to either $8$ or $9$. However, fortunately for finding a solution, they are consistent in that, considered to their greatest common divisors, the relevant pairs give the same constraint:
$$\begin{align}
\left . \begin{array}{l} x\equiv 4 \bmod 6\\[1ex] x \equiv 4 \bmod 9 \end{array} \right\} & \implies x\equiv 4\equiv 1 \bmod 3\\[2ex]
\left . \begin{array}{l} x\equiv 4 \bmod 6\\[1ex] x \equiv 6 \bmod 8 \end{array} \right\} & \implies x\equiv 2\equiv 0 \bmod 2\\
\end{align}$$
Reassured of their consistency, the first pairing immediately gives the constraint to their least common multiple:
$$\left . \begin{array}{l} x\equiv 4 \bmod 6\\[1ex] x \equiv 4 \bmod 9 \end{array} \right\} \implies x\equiv 4\bmod 18$$
and a short search turns up the appropriate value to incorporate the $\bmod 8$ constraint also:
$$\left . \begin{array}{l} x\equiv 4 \bmod 18\\[1ex] x \equiv 6 \bmod 8 \end{array} \right\} \implies x\equiv 22 \bmod 72
$$
then adding in the $\bmod 5$ constraint we find
$$\left . \begin{array}{l} x\equiv 3 \bmod 5\\[1ex] x \equiv 22 \bmod 72 \end{array} \right\} \implies x\equiv 238 \bmod 360
$$
And finally the $\bmod 7$ constraint gives
$$\left . \begin{array}{l} x\equiv 0 \bmod 7\\[1ex] x \equiv 238 \bmod 360 \end{array} \right\} \implies x\equiv 238 \bmod 2520
$$
Thus the (positive) numbers satisfying the first four constraints begin with $238, 598, 958, ...$ but those satisfying all constraints start $238, 2758, 5278, ...$
