The "sex game gone wrong" theory assumes facts not in evidence.......
With regard to pulling the door shut, I imagine that there may be a knob/handle of some sort further towards the centre of the door, which isn't visible in these photos. Otherwise, as you say, it would be very difficult indeed for anyone to pull the door closed from the outside. But it's equally clear that there's no turnable lever handle on the outside face of the door.
Re the wood that appears to be jamming the spring latch open: I have a fair idea. It's pretty clear that the spring latch would only be operable via the inside lever handle. Therefore, if it were allowed to latch shut, nobody would ever be able to enter the front door from the outside without assistance from someone who was already inside - regardless of whether the cylinder lock below was used or not.
So, it looks to me as if the person who installed the door furniture quickly realised he'd made a stupid mistake - i.e. without a corresponding lever handle on the exterior face of the door, nobody would ever be able to open the door from the outside without someone on the inside turning the inside lever handle to open the spring latch. And for this reason (I believe), the spring latch was wedged permanently open.
Good explanation?
Not really.
I've seen just about every screw-up in installing door hardware that can be envisioned by the human mind, and more than a few that hadn't been, at least until they happened.
I've never seen anyone accidentally purchase and install a residential entry door mortise lockset and not notice that there was only one handle to the assembly until they were finished.
There have been many rather dubious stretches of probability to account for things in this thread, but that one is almost in a class by itself.
Note that the face of the door without the handle is an unblemished surface. This suggests that either the door leaf was factory prepped to accept a lock assembly with only one handle, or that the installer did the prep. I can say with great certainty that anyone who is competent to prep a blank door leaf for a mortise lockset is very unlikely to be as incompetent as you surmise. It requires a high level of skill and ability, a fair amount of practice, and a deep familiarity with both the hardware being installed and a good number of tools.
I spent a little time trying to track down that particular model of lockset. I'm pretty familiar with most of Corbin's products, but I haven't run into that one, and I haven't found it on the web yet. It may be one that is in more general use in Europe, or perhaps just Italy, but not the U.S.
What I was trying to find out is if the spring latch was operated in conjunction with the deadbolt from the outside cylinder. This could be a security function designed to compel 'key entry only' from the outside, but obviously the inconvenience would prompt users to defeat the system in just such a fashion as we are seeing after they locked themselves out a few times.
This is not a very appropriate configuration for a residential entry door. In the past a similar sort of configuration, albeit with handles on both sides, the pull side inoperable, (one of the permutations of a "storeroom" lockset) could be found on some mechanical room doors, most often in buildings with rooftop mechanical rooms which opened onto an accessible roof. They had less in the way of such lock-out issues because the people who went out onto the roofs through those rooms were usually service personnel with their keys strapped to their belts, but they have been discouraged by most modern fire codes.
The thing is, if that is the state that the spring latch was in when the crime occurred it would mean that there isn't any 'little bit' or 'occasional' about the door not latching. It would
never latch, at all, and that would probably be quite obvious to anyone using the door. At best it might 'snug' shut by friction against the frame, but we have reason to believe such friction would be minimal, since we have been told that only a breeze was needed to swing it open.