Actually no. Radiant cooling is the bulk of human cooling.
Convection? Well indirectly it may play a role - it brings fresh cooler air next to the body to conduct more heat away, and fresh drier air to evaporate more away. But not as a direct means itself and not a major factor compared to radiant losses.
Under ideal circumstances - a person at rest, buck-nekkid, glabrous, with arms legs and butt-cheeks spread out to obtain optimal view factors for radiative transfer - I suppose this is true.
I was assuming a human bean engaging in aerobic exercise, say a quarter of a horsepower mechanical power output, or 186 watts. The conversion efficiency of muscles is about 25%, so for 186 watts of mechanical power output, the body has to dump about 550 watts of waste thermal power.
If you’ve got head/body hair, shorts, socks, and shoes, and maybe even a T-shirt, you will substantially limit radiative losses, even as skin temp creeps up. From your link, I estimated an exposed skin area of 1.5 m^2, and a peak skin temp of 40 Celsius, giving a radiative loss of 160 watts. You’re right, it’s not neglible, but for a person exercising hard, it’s not the bulk of their cooling.
It’s not clear what you mean by the direct/indirect descriptors. You’ve described exactly how convective cooling works, i.e. by bringing fresh cooler air next ot the body to conduct more heat away; there is no more direct way for convective cooling to work. 
There is this little bit from your cite:
http://hyperphysics.phy-astr.gsu.edu/HBASE/thermo/sweat.html#c1
So what conditions are we talking about? An air-conditioned gym, or a hot summer day outside?
When ambient temperature is below body temperature, which averages 98.6 but is a bit higher when working out and in need of cooling, so say under 100. Which includes most hot summer days.
Of course clothing limits conductive losses and evaporative losses as well.
By direct/indirect descriptors I mean that you have to classify a loss as one class or another. The actual loss is conductive or evaporative and you can’t count the loss twice even though convection increases the efficiency of those classes. Direct convection heat loss is when hot air rises out of a room and cool air flows in resulting in a temperature lowering. That does not apply to body cooling.
In any case the question was why (if indeed it is true) wiping off heavy sweat makes us cool faster than leaving it on. My answer was that the evaporative loss was no greater with heavy sweat than with a thin layer but that heavy sweat functions as an insulating layer. That would be true if the insulation is preventing radiant or conductive losses of course; my source specifies that radiant loss is usually the bigger factor.