Any rocket or bullet turned against Israel for the past 20 years has its roots in Iran. Iran controlled Lebanon, Syria, Yemen, Gaza and Iraq through its militias.
This rhetoric reminds me of how, at one time, everything bad that happened to Germany was said to be orchestrated by you know who.
Those who shall not be named did not fund, arm, train and direct militias that controlled Austria, France and Poland.
German media said they did
Germans believed they did
If you knew anything about how Iran is doing this, you'd know they are happy to throw money or materiel at anyone who can put together a force against Israel, or even Serbia during the Bosnian war. They hardly direct or control anything afield, other than being mostly ideologically aligned. They don't need to pull puppet strings like you say. They happily work with resistance forces regardless of sect.