Proving that if $L=\{ a^n b^n c^n \colon n\ge 0 \}$ than $L\notin CFL$
Frankly, I think your argument works. As there is always one of the letters missing, the second condition forces the substring to be "short", pumping will not get equal amounts of $a,b,c$.

For didactical purposes authors in books take extra precautions to make sure the student understands that all cases must be covered. Hence they tend to avoid shortcuts.