Awesome question - and awesome observation! Yeah, the latch needs to click/lock into place in order for the car to agree to start charging.
The latch tip tends to get bent sometimes, especially if it lands on the floor tip-down (look for a little divot in the tip of the latch). The latch is made of aluminum, I believe, which can be brittle if bent too much, and when it starts bending, it bends rapidly - so apply careful calculated force to un-bend it.
I find it easiest to unbend the ITT-manufactured handle latches using an adjustable wrench, sized to perfectly "wrap" the latch tip (straight-in, not trying to grip the latch face at a 90-degree angle which is impossible). When it's on, you can use the wrench handle to carefully bend the latch straight with calculated force.
It should be bent such that, when you insert it in the car port, the latch falls into place just at the end of the plug's travel - no need to push the handle harder into the socket, but also not loosely latching too early either.
If you can, a photo would help diagnose further if needed - but I hope this helps!