Why are quotient metric spaces defined this way?

This is to ensure the triangle inequality. In your proposal, it can happen that $[p]$ and $[q]$ have nearby representatives and $[q]$ and $[r]$ have nearby representatives, but the two representatives of $[q]$ involved are different, so this doesn't guarantee that $[p]$ and $[r]$ have nearby representatives, so the triangle inequality may be violated.

With the Wikipedia definition, on the other hand, it's straightforward to verify the triangle inequality, since any chain of points from $[p]$ to $[q]$ and any chain of points from $[q]$ to $[r]$ can be concatenated to form a chain of points from $[p]$ to $[r]$, so the triangle inequality follows from the individual triangle inequalities. In particular, in the above situation, we can "hop" from one representative of $[q]$ to the other without extra cost.

Clearly, using the definition from Wikipedia, you always get a smaller or equal distance than under your proposal. If the definitions disagree and your approach is employed, the triangle inequality is violated. The definition you find in Wikipedia guarantees that the triangle inequality is valid. So I will give an example in which the triangle inequality is violated under your propsal.

Let $A=\{(0,x):x\in[0,1]\}$, $C=\{(1,x):x\in[0,1]\}$, and $B=\{(x,x):x\in(0,1)\}$. Let $X=A\cup B\cup C$ and endow it with the Euclidean metric. Partition $X$ into $A$, $B$, and $C$. One can make a path from $A$ to $C$ have lenght arbitrarily close to $0$, but $d(A,C)=1$ under your proposal.