Silicon and germanium are in the same column of the periodic table, and they're both semiconductors, so I think this could certainly be made to work. Actually, silicon is the element more commonly used in electronics. The manufacturing processes for producing bars of high-purity germanium are basically just an offshoot of the processes used with silicon. If it weren't for Silicon Valley, we probably wouldn't be able to make HPGe detectors.
Will it work when cooled to the same 77K?
There's nothing magic about 77 K. That just happens to be the boiling point of nitrogen, which we use as a cheap and convenient coolant. It's not as though an HPGe detector won't work at 78 K. I believe that at higher temperatures, you simply wouldn't get such good energy resolution, and if you try to operate it at very high temperatures, such as room temperature, it will damage the detector. I don't know for sure about silicon, but I imagine that due to its similar chemical properties, the operating range would probably be about the same.
Why HPGe is mainly used for gamma spectroscopy then and what are the advantages vs Silicon?
Germanium is atomic number 32, compared to silicon's 14, and it also has a higher density of electrons. These properties greatly increase its efficiency for detecting gamma rays for a detector of fixed volume, and also increase the probability of getting the full energy deposited in the detector, rather than getting only part of the energy in a Compton scattering event in which the scattered gamma escapes.