What is the design philosophy behind the moment-based quantum circuit? What are the advantages and disadvantages of it?
The basic idea is that we wanted to give users more control over what will actually happen on hardware. Whether or not two gates are run in parallel is really important information when dealing with noise (e.g. it determines total runtime), so it's important for users to be able to determine-whether that's happening and also control-if that's happening.
The philosophy of giving more control isn't limited to moments. Cirq also requires users to specify specific qubits (no implicit placement), and to use the native gateset of whatever device they are targeting (no implicit decomposition). The important thing is that you should be able to have a clear concrete idea of what will actually happen on the device.
Cirq also has the concept of a "schedule", which has even more control over timing. But ironically we've found that users don't want to use it, and programmers don't like consuming it. So we're actually cutting that next version.
The list-of-moments structure seems to be a nice sweet spot in terms of flexibility, ease of production, and ease of consumption. You can certainly put the abstraction boundary in other places, but that's where we've found ourselves pulled towards.
It seems to me the earliest insert strategy is always preferred
It's a good rule of thumb, which is why it's the default. But it's not always optimal. For example, hardware may perform best when circuits are stratified into layers of single-qubit gates and layers of two-qubit gates (but never a layer with both single and two qubit gates). So even if you can move a single qubit operation earlier, that may result in it being in the same layer as two qubit operations. Moving it would not be a fatal mistake, the circuit would still run, but it might be slightly noisier. There are lots of "little things" like that when dealing with hardware.
