The theorety is that if he got enough legal ammo (and a judgement asserting that Wright is Satoshi would be part of that) he'd be able to force most/all Bitcoin Core devs to push for this change. My understanding is that his legal harassment of Bitcoin Core devs was part of this plan.

Of course in reality it would just lead to a hard fork and sow mayhem in the Bitcoin ecosystem.

It could have some highly amusing consequences if serious legal pressure was put on bitcoin. It could split bitcoin in half, with miners and users rejecting the pressure and picking one side of the fork, and core devs, mainstream exchanges, and hedge funds being vulnerable to the pressure ending up on the other half. The immediate consequence might be hundreds of billions in value evaporating, but if the legally approved fork won, it could lead to bitcoin going all the way to 0 in the long-run, and being replaced by another currency.