Maybe this is just my "hope" talking, however, from the start I've felt that he left home and left his cell phone there, as if leaving "everything" behind him. I think Liam ran away, left home.
Did Liam go to someone's house to get away? Is this someone his parents aren't aware of?
The fact that he didn't say where he was going is puzzling.
Thanks for letting us know that the doors Liam walked through are the entrance to "Union Square". Now I have an even better picture of the area there.
You are absolutely right - we have no CCTV of Liam either boarding or sitting on the bus.
As to when his family saw him last, the police state that Liam was seen on Saturday afternoon. This would be the CCTV at McDonalds. No mention, as you say, of the last time his mother saw him. I find this puzzling, as well.
The main reason I can think of as to why a person of that age would leave a cell phone behind is that he or she would know that they could be "tracked" on it.
The police and his mother have been urging anyone to contact them to tell them at least that Liam is safe. If there was a reason why Liam left, only his mother and the police are privy to that information. That is - if they know.
As you say, Liam's state of mind has not been mentioned.
This case has tugged at my heart since day one.
Whatever the scenario turns out to be, I still have hope that Liam is safe somewhere. There is always hope.