Kylotan said:
some part of the system is not sending important updates quickly enough
It's true, you should aim for the highest possible packet rate for control inputs for players who are visible. 60 Hz packets are totally possible on most current internet connections.
That being said – latency is latency. If you forward extrapolate, you will absolutely overshoot when players make drastic control changes. The options are to either try smoothing the overshoot, or to not forward extrapolate. Which variant works best, depends on the specific game.