Can we find a bound so that we can conclude $G$ is a $p$-group?
Solution 1:
Yes such general bound actually does exist. It was proved by Thomas J. Laffey in "The number of solutions of $x^p=1$ in finite groups". That theorem states:
If $n_p > \frac{p}{p + 1} |G|$ then $G$ is a $p$-group
Note, that for $p = 2$ this theorem yields a result, stronger, than yours (namely $\frac{2}{3}$ instead of $\frac{3}{4}$)