I agree.
Yes, there are a lot of factors that contribute to the heat gain, but, most are not as significant as the friction factor or the gas temp on the base of the bullet. Things like the compression of air in the bore and the air friction outside the bore would serve to keep the bullet hot after it left the barrel, maybe even increase the temp slightly.
The interest here is why is the lead melting under the gas check and not on the sides of the bullet? Maybe a couple of reasons. 1) the gas check is copper and a good head conductor, which means the gas temp on the base of the bullet is 'felt' by the lead. 2) the friction of the gas check is higher than the lead portion of the bullet. 3) lube on the lead portion of the bullet is reducing further the coeff of friction of the lead portion of the bullet.
Another data point. I have found when sizing bullets that the GC has a slightly larger dia (~0.0005) than the body of the bullet. The simple explanation is the copper GC has a 'spring back' and the lead has less. Higher yield limit maybe the source? Is this related to how the GC acts in the barrel or why it is heated more?