Combustion of saturated hydrocarbons vs combustion of unsaturated hydrocarbons
You are viewing it from the wrong perspective.
In the case of normal burning, Generally, the oxygen is the limiting reagent (therefore some of the hydrocarbons left behind ).
Let's take 1 mole of Oxygen.
So you have to take  0.285 mol of ethane (0.57 mol carbon )  and 0.4 mol of ethyne (0.8 mol of carbon).
It's quite clear ethyne has more carbon content for a given amount of oxygen. As if some hydrocarbon get left, Ethyne will give more carbon content.

https://www.youtube.com/watch?v=FJi4ODPQSE0

(Note: If concentration of oxygen is very high and enough temperature is given then both will burn with Blue Flame, as it is done in case of welding
https://www.youtube.com/watch?v=hKMDxag8yAo )

Now In the case of welding, welding is done in presence of excess of Oxygen, so there is no question of limiting reagent of oxygen.
What matters is Auto Ignition point, Ethyne has much lower auto ignition temperature that is 305 °C as compared to ethane which has 515 °C.

Of course, you pointed out above that for, given 1 mol of ethyne and ethane, ethyne need less oxygen to be burned, so it can be taken in quite concentrated form.

You can take maximum 80% Ethyne (concentration in air), it will burn.
Whereas ethane can't be burn as its concentration exceeds 12.5% (in air). 

More concentrated heat implies Higher can be temperature achieved.