@MelmothTheLost as you are a volunteer investigator with Locate International, I would be very interested in your thoughts about how you think the investigation into finding Gus has gone so far and would you have done anything differently.
With your Locate experience and being on here for 9 years I thought you may have some useful insight, experience or knowledge of carrying out a search to find a missing 4 year old such as Gus. Only if you want to, of course
I realise I didn't address all of your questions.
From my (arm's length) experience, LE would have initially taken the family's report at face value on the basis that a young child was missing and therefore presumed to be in danger. The immediate focus would therefore be on the search that we have all been following for the past couple of weeks. I would expect them to ask about such things as mine shafts and abandoned wells and any other hazards where a child could get trapped, and for the family, having owned the property for well over 100 years, to know about many, if not all, of them and that known shafts and wells etc would be included in searches.
In the background, they would want to rule out parental abduction, especially in view of Josh's poor relationship with Josie and his apparent disagreement with some aspect of how or where Gus was being raised. There would therefore be enquiries into Josh's whereabouts at the time Gus was reported to have gone missing. He may or may not have been definitively ruled out on the basis of those enquiries.
I would expect there to be repeated interviews with the other adults at the property, with the focus possibly changing as the days pass and no sign is found of Gus. We know there were three adults at the property - Gus's mother, his grandmother and Josie. Do their accounts remain unchanged or do details change as time passes? And yes - there would be background checks on all adults on the property for any criminal records or other reports of interest.
There would also be questions about the nearby properties, who owned them, who was living in them and whether they were occupied at the time Gus went missing. I don't recall reading whether these were independently owned or part of the station, eg accommodation for station hands.
And visitors to the property, including (especially?) the ones who become invisible because of their familiarity, such as the postman or Amazon delivery person who visits several times a week.