You gentlemen are overlooking an important mode of melting the ice/snow inside the probe.
Infrared radiation. Some initial reading on the subject tells me that we are lining the interior of the probes with the wrong substance. From a Nippon Steel technical report on infrared spectroscopy, this telling observation:
http://www.nsc.co.jp/en/tech/report/pdf/n10014.pdf
The emission energies of the glass sheet and PTFE sheet at 200 Celsius
are so high that both materials radiate heat when you draw your
hand near towards them. By contrast, none of the metallic materials
feels hot.
I can think of geometries that would promote both modes of melting (conduction and radiation) and which would promote development of open channels through an ice mass.
Seems to me that pitot tube heating design is stuck in the 1930's.