Thinking gradually as you said I could especulate something. Birds already have to get trought a strenuous process to extract themselves from an unburied egg. One could imagine the early stages of the evolution of egg burying involving a very shallow layer of dirt that way, so the diference wouldn't be so sudden.
Now since there are evolutionary advantages for deeper burial (I think). Then there is selective pressure for greater depth of burial. So after many generations of slightly increase of burial depth, the individuals who had slightly increase of physical activity to reach the surface would have a higher chance of surviving, thus having a higher chance to leave offspring.
Well, this certainly sounds like a plausible explenation, I think you just answered your own question.. With plausible I mean that it could be some of the ways that would give the birds an advantage over those who don't bury their eggs.
You are correct in stating that gradually exposing its eggs and offspring to more depth would give certain offspring a bigger advantage if those offspring had a mutation that managed to exploit it.
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