Why would a thin speaker wire be a problem? An undersized wire would be higher resistance than a properly sized wire for the amount of current going through it, and a higher resistance presents less of a load to the head unit. So I'm not buying that at all. Even the small increase in inductance wouldn't bother the head. The power wires to the head unit are a different story, they could be undersized pretty easily. The voltage drop across a grossly undersized power or ground wire could possibly make the head shut off, sure. With 62x4 w peak, the head could draw upwards of 20A very briefly, which is a bit more than a 20 or 22 gauge (common sizes for stock power wires) would handle without sizeable voltage drop across it.
I don't think the speaker wires/outputs are the issue at all though, they are only 62x4 peak, so their continous (average) power is much lower than that. That's nothing compared to the 800w average rating of the amp.