If the case is not solved it would be hard to justify turning a review into an investigation in my opinion. Surely an investigation would only be carried out if all concerned believed it would lead to a solution? If it doesn't then I would like to know why it didn't.
Hmm.
There was a scoping exercise, eventually followed by a review with an investigative element, then the Met decided that they had enough elements to open a full investigation (as well as appeal for further information). Having all the bits and pieces on a database would presumably have thrown up all kinds of red flags - ranging from the potentially suspicious to loose ends to tie up once and for all and eliminate the irrelevant. Aside from Madeleine, there were also other British children, some of whom had been sexually abused and also deserve an attempt at justice, whether there is a link or not.
IMO, as more and more of the plod work gets done, the Met may well cut down on the staff involved and an associated drop in the allocated budget. If that happens, the Met will probably make an anncouncement in the media to that effect, stating that they believe that they've investigated as thoroughly as possible (x "actionables", etc., have been pursued), and operations will wind down until there is a major breakthrough.
I don't see what would require a public inquiry... even the Jersey fiasco wasn't a public inquiry, but an independent police one.