The laptops I had the problem with were both Toshiba with original PSU.
I never sorted out exactly what was going on as these were used by other people, so I couldn't spend too much time 'playing' with them.
There may have been faults in the amps they were connected to or some effect of the long spur wiring in the locations where they were used, but running on battery produced a big improvement. As they were only occasionally used for audio, that was sufficient to get the problem off my list!
I'm not sure which brands should be reliable these days. You may remember my problem with a set of leads on HP laptops going high leakage and causing the RCCB to trip.
The point about 'genuine manufacturers' PSU' is well made though. I bought a replacement Dell PSU on ebay last year, which looked pretty convincing in the photo, though the price betrayed it. I took a chance and sure enough it went bang within a week. When I took it apart, I was really amazed at the appalling quality of build and components. Though I didn't bother to analyse the circuit, there was a lot less than inside good PSUs I've taken apart.
When I used the scope/spectrum analyser part of the program, I was just using a microphone to display sound waveforms, so no risk there. If I were going to probe around in circuits, even low voltage battery powered, I think I'd build some sort of front end. Aside from protection, it would be good to get the input impedance up to the standard 1MOhm or more. There are some around on the net I think.
Incidentally, back to the OP, my audio sig gen (actually Unilab, not Griifin as originally posted) is essentially a function generator chip and a beefy amplifier chip. So the answer I think is, yes, a function generator chip could be used. Mine goes up to 100kHz whereas the soundcard generators would be limited to about 20kHz I guess.