If the training is being done in real IMC not checking the instruments and aids is simply unacceptable. I do not see how doing the radio aids checks from the start of the course will overload the student, as the aircraft is not flying. I think that some instructors just want to get flying and therefore earning as quickly as possible.
Not sure if it has been mentioned already but the OAT gauge should be cross checked against the ATIS.
The GPS checks should include checking the satellite status page ensuring the EPE is less than 50ft. DOP less than 4, if HUL is calculated less than 0.3nm. On the Garmin 430/530 the first test page should be cross-checked against the CDI or HSI presentation to verify the halfscale deflections, the annunciator lights and that the bearing is consistent with that selected and changes with the OBS.
I have encountered IMC rated pilots who do not know what a Radar Vectored ILS is and did not know what circle to land procedures were. I also know of instructors qualified to teach the IMC rating who will not fly in IMC.