How would an EV know that the latch on the plug has locked on to the charge port's locking pin? The locking pin has no sensors on it so doesn't know whether anything has latched to it. There's no sensor on the latch, so the EVSE doesn't know that the latch is broken.
The button on the plug that lifts the latch is attached to a switch that tells the EVSE when the latch button has been pressed or when the latch has been lifted by the charge port locking pin as the plug is inserted into the port. When either happens, the EVSE turns off power to the plug pins so that arcing doesn't occur when the plug is inserted or pulled out of the port. When the button is released or the latch has slid over the locking pin and falls down into the locking pin to secure the plug, the EVSE powers the plug pins.
When the latch is broken off the plug, the plug could be inserted into or pulled out of the port while the EVSE is still powering the plug's pins resulting in arcing. It would be better to press the latch button to shut off the power to the plug's pins to prevent arcing, but not doing so shouldn't prevent charging from succeeding. Maybe in some EVSE's, charging won't start until the locking lever has lifted and the dropped as would happen when the plug is inserted with an unbroken latch. If so, just pressing and releasing the latch button should power the plug's pins.
Searching for "EV charging with a broken latch" finds several accounts of EV owners who've successfully charged using plugs with broken latches as well as others who've not been successful. I might have been successful charging our Mitsubishi i-MiEV with a broken plug latch more than a decade ago and might not have had this experience with our i3's, so maybe there's something that prevents an i3 from charging when the plug latch is broken.