What is the problem with this method while integrating $(e^x-(2x+3)^4)^3$?

Your methodology is highly nonstandard and suspicious, but I will try to formalize it.

You appear to deal with integrals of the form $$ int f(x)^n dx $$

I think your general strategy is

1. Set $y= f(x)$

2. Integrate $y^n$ to get $y^{n+1}/(n+1)$.

3. Divide $y^{n+1}/(n+1)$ by $dy$ and then write everything in terms of $x$.

So in the end you are saying that the following is an antiderivative for $f(x)^n$ $$ frac{f(x)^{n+1}}{(n+1) f'(x)}tag{1} $$

So let's take the derivative of the last function and see what we get (we should get $f(x)^n$ if your strategy is correct). Applying the quotient rule and chain rule, we get the derivative of the function in $(1)$ as:

$$ frac{f(x)^n (f'(x))^2-frac{1}{n+1}f(x)^{n+1} f''(x)}{(f'(x))^2}tag{2} $$

In general, there is no reason to expect that the function in $(2)$ is going to equal $f(x)^n$.

On the other hand, if we are in the special case that $f'(x)$ is a nonzero constant, then the expression in $(2)$ is equal to $f(x)^n$ since in this case $f''(x)=0$. This is precisely what happens in your two easier examples where the strategy works. But this is not the case in the more complicated example.