Glenn Tattersall and colleagues published infrared thermography of toco toucans in Science in 2009. The bill has a dense superficial vascular network beneath a thin keratin surface, and the birds regulate flow through it. Bill surface temperature swung rapidly with ambient conditions and with activity, and the rate of heat loss through the bill varied several-fold within minutes.
At the highest measured rates the bill accounted for a large fraction of total heat loss, far more than its share of surface area would suggest. At night, toucans tuck the bill under the wing and beneath the tail and flow drops, conserving heat. The structure is therefore a controllable radiator rather than a fixed liability.
The older explanations have not vanished, and need not. The bill is also used to reach fruit from thin branches, to raid nests, and in social display. Thermoregulation is an additional function demonstrated by measurement, not a replacement for the others.