Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?

The assertion is incorrect. $L^p$-convergence (for $p < \infty$) does not imply pointwise convergence even for compactly supported smooth functions.

An easy example is a shrinking bump function, if $f \in C_c^\infty(\mathbb{R}^n)$ has $f(0) \neq 0$, then the sequence $(f_j)$ where $f_j(x) = f(j\cdot x)$ converges to $0$ in $L^p$, but $f_j(0) = f(0)$ for all $j$.

You don't even have pointwise convergence almost everywhere, as was the case in the above example. If you take a compactly supported function $f$ that takes the value $1$ on the entire unit hypercube (whichever dimension the space has), and then shrink the support by scaling with $\frac1k$, then moving around the scaled function so the value-$1$ plateau covers the entire hypercube (thus you first have $2^d$ translates of the scaled-by-$\frac12$ function, followed by $3^d$ translates of the scaled-by-$\frac13$ function, then $4^d$ translates ...) before the next shrinking, you have a sequence that does converge to $0$ in $L^p$, but not in any point of the unit hypercube.

However, $L^p$ convergence always (smooth or not) implies the existence of a subsequence that converges pointwise almost everywhere to the $L^p$ limit function.