To make stability proofs simpler, I can consider an explicit scheme written as $$V(n+1,i)=aV(n,i-1)+bV(n,i)+cV(n,i+1)$$ and one can show that if $a,b,c\ge 0$ and $a+b+c\le1$, then the explicit method is stable.
And for implicit, the scheme can be written as $$aV(n+1,i-1)+bV(n+1,i)+cV(n+1,i+1)=V(n,i)$$ and if $a,c \le 0, b\ge 0$, and $a+b+c>0$, then the scheme is stable (This condition can be found in Smith book on ths stability of Thomas algorithm).
However, all the finite difference books typically talk about Matrix stability or Fourier stability. But what I have outlined above is not PDE dependent, so why could not I just check the test on coefficients as above and never learn about other stability methods? Does this rule has limitations?