Reinitializing Subscription Performance
It really depends on how old the snapshot is.  After the snapshot is applied, the agent will need to also send all of the changes that have occurred since the snapshot was taken.  Depending on how many changes have occurred, this may or may not affect other existing subscribers.

I tend to take a new snapshot if possible before reinitializing a subscriber although sometimes that is not an option.

I wouldn't worry too much about it impacting other subscribers.