but shy of first place

[rainous]

Melbourne has scored near the top of global livability surveys for the past decade but shy of first place.

Is it correct to assume that something been omitted in front of "shy of first place" and if so, could you tell me what it is that has been omitted?

[Tdol]

It just means that it has come very close but has not reached the first place.

[Raymott]

Melbourne has scored near the top of global livability surveys for the past decade but shy of first place.

Is it correct to assume that something been omitted in front of "shy of first place" and if so, could you tell me what it is that has been omitted?

The meaning is, "but has always been shy of first place". So yes, you could say, "has always been" has been omitted.
Even so, "near the top, but shy of first place" makes sense on its own - in which case, nothing has been omitted; it was just never there.