This DOES seem like a simple implementation. Just like period offset, except can be set anytime instead of only the initial starting point.
Worry not, for I've found a nice solution if you want to restart the sine!
Setting the Sine period to 0 will always keep the cycle position at 0, so you can abuse that to quickly reset the sine to the start.
However, it's still not as fluid as a Set Period option. I'm honestly surprised it hasn't been done already. Why not, guys?