Violent engine shaking at startup. Miss fire codes all cylinders

That's good. There's a "rumor", that there's a corellation between clogged EGR and head gasket failure. The EGR gasses moderate combustion chamber temperatures, and the way the temperature rises/falls during the combustion process.

There was one report here, someone replaced head gasket without cleaning the EGR, and the new head gasket failed withing another 20K miles.

One thing more: did you also clean the intake manifold thoroughly? There are EGR passages in the IM. The EGR gas feeds into a large passageway on the side of IM, below and to the right (as you stand at front of engine bay).. That passageway splits up into four capillary passages, one at each intake port. Those capillary passages are particularly clog-prone.