I'll stick my head above the parapet and say, whilst everything you say is a possible explanation of the events it is just surmise and genealogists aren't supposed to make unsupported suppositions.
One thing that sticks out to me is that the whole thing relies on accepting the accuracy of what was said about Kathleen's birth, including dates etc.
I understood that not registering a birth was quite a serious offence and would have required the connivance of several authorities (mid-wives, doctors etc.) or by the parent(s) moving around a lot. So I would look to see if I could find a suitable birth in wider time parameters and possibly registered to other family members, occasionally a birth would be registered to a mother or sister of the actual mother to avoid scandal.
Is it possible that the trigger for George's 'insanity' was discovering his wife's infidelity? Actually it's unusual to die of 'insanity', that was possibly used to cover other things, STD, suicide etc. Remember at that time unmarried mothers were still being placed in asylums simply for being pregnant. But there I go, committing the sin of supposition.
Currently investigating the Hillmans of Sussex.