Yes the do investigate, if the family lodges a formal missing persons report - seems odd that the partner claims to have reported direct to the RTP though, as she is in the UK. Normal process is for your family to report to local police (ie: she would report in the UK) then they pass it into Interpol, which pushes it into the RTP system. If requested by the family the Embassy could ask for update from RTP, but consular staff aren't law enforcement and don't have any remit to conduct searches in Thailand.
There's always a chance he's been arrested or hospitalised and has withheld permission for his family to be advised of his situation. That certainly happens, but in most cases the Thai media eventually gets hold of the information and the families find out anyway.