> it is reasonable to assume that the secondary battery was fully charged before descent

Actually, no, it's not, unless you show something to prove that. Have you ever designed the device with two different power sources supposed to work on a comet after travelling 10 years through the space and being launched from the orbiter? I haven't, and can't claim it's reasonable to assume, whereas being an engineer I can imagine a lot of reasons for the directly opposite claim. What are your experiences or citations that allow you to claim that? Please state them or avoid uninformed claims.

Specifically, if the secondary batteries were supposed to be fully charged before, why would the non-rechargeable batteries be the primary batteries at all?

And technically, there are more primary batteries and more secondary batteries in Philae, the primary batteries having 1000 Wh, the secondary batteries having 140 Wh. Note that if you use e.g. 30 W during experiments, you use the primary batteries in some 30 hours. The scientists had to decide the rate of the battery use. Just like your mobile phone lasts only hours if you speak all the time on it. The lander being in the shade had different temperature environment than in the ideal case. It's extremely cold there and the high enough temperature is necessary for the things to work.

Compare all that with your modem constantly using 10 W, your computer, if notebook, using probably 20 W for serious work. Even without the heating problem, you wouldn't be able to use internet and work for more than 30 hours with Philae's primary batteries, and certainly not for 4 days.