It's hard to blame Iran when they literally weren't doing anything until we attacked them. You don't blame a child for punching back against the school bully - unless you're also a bully.
I don't think Iran are entirely blameless given their sponsorship of Hezbollah and participation in the Yemeni (and previously Lebanese!) civil wars. They've been maintaining a sort of sub-threshold threat in the area for a long time. Which was why, yes, it was an incredibly bad idea for US/Israeli attacks to escalate that.
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.
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.
