Good Afternoon,
These posts on the flap closure and retention often drive me round the bend. It's the words 'latch' and 'catch' which are used in abundance, some authors seems to start off with one meaning, mix them up randomly and I never sure which bit they are talking about, or it me just being thick. This does not apply to you, you are clear with an excellent description of your investigation, thank you.
Having just spent some time looking at various websites on locks, difficult to find a definition of 'latch', and a million and one types of lock complicates, but what is a strong theme is the term 'latch' is the bit that is visible with door/flap in our case is open. Its function is to allow temporary retention of the door/flap in a closed position, moves in and out and can be tested with say a finger to check smooth action. It does seem to be a contraction of the term 'latch bolt', the bolt being inside a lock but to be clear I will mean a 'latch' is the just the bit you can see when open. To try and be clear some diagrams...
View attachment 102067 View attachment 102069 View attachment 102070
In the context of the A2 flap, to be clear, this bit.
View attachment 102071
(Rant over.)
To eventually get to the point, it is your words 'pulling out' in the sentence 'Ive tried pushing the latching piece in and pulling out and it seems to move freely.' that worries me. The whole point is should not need pulling out, like many latches it is spring driven, it should ping out when your finger is removed from the pushed aside state. Further without the encouragement to remain extended it will be loose with the catch and not retain the flap securely. (I use the term 'catch' to be the bit attached to the flap, bottom left in the last picture, that engages with latch.)
Notice George's reference to WD40. The best thing from memory is remove the latch from its fixing, couple of screws if I remember correctly, which will allow a vertical orientation so the WD 40 can get in properly.
Sorry for the length and rambling but hope something might inspire.
Andy