Why aren't all functions considered a power function of 1 and integrated using some sort of chain rule?

You forget the chain rule. Indeed,

$$\frac d{dx}\frac{y^2}2=y\frac{dy}{dx}\ne y$$

Which is why integration is so much harder than differentiation.

However, it is true that

$$\int[f(x)]^nf'(x)\ dx=\frac{[f(x)]^{n+1}}{n+1}+c$$

Which should follow from u-substitution or differentiating both sides.

The power rule for integration says that $$ \int x^n dx = \frac{x^{n+1}}{n+1} $$ Why doesn't that work for $\int [f(x)]^n dx$? Because $f(x)$ does not match $dx$. The proper use of the power rule would be $$ \int f(x)^n df(x) = \frac{f(x)^{n+1}}{n+1} $$ But we can still get something useful out of this by using, as you put it, some sort of chain rule. Namely, the chain rule, which tells us $df(x) = f'(x) dx$, which makes the above into $$ \int f(x)^n f'(x)dx = \frac{f(x)^{n+1}}{n+1} $$ As you may have noticed, this is just a $u$-substitution with $u = f(x)$. In fact, $u$-substitution is in general equivalent to the chain rule, and thus is the sort of chain rule you describe in the question.

It is wrong. The derivative of $[f(x)]^2/2$ requires you to use the chain rule, so $$\frac{d}{dx}\frac{[f(x)]^2}{2}=\frac{2[f(x)]^1}{2}f'(x)=f(x)f'(x).$$