Why are acetic acid and sodium bicarbonate mixed to unblock a drain?
Quoting text from this source under the section &quot;Why could baking soda and vinegar clean clogged drains?&quot; :

Bicarb soda and vinegar react because of the acid-base reaction. Bicarb soda is bicarbonate $(\ce{NaHCO3})$ and vinegar is acetic acid $(\ce{CH3COOH})$.


When bicarb soda and vinegar react, they fizzle and sizzle and they expand. This is why they are able to remove clogs from drains. The pressure from the expanding product shoves the clog down as it moves along. The vinegar solution (vinegar plus water) can remove soap residue that clings to the walls of the pipes.

So it seems like your conjecture is justified, that it is not actually the reactivity of these two substances with the waste that is responsible for majority of the cleansing, but simply the mechanical action of the expansion of the carbon dioxide gas and the forming product that pushes the waste down the drain, and leads to unclogging.
The relevant chemical equation is:
$$\ce{CH3COOH (aq) + NaHCO3 (s) -&gt; CH3COONa (aq) + CO2 (g) + H2O (l)}$$
This is similar to the mechanism by which baking powder makes a cake fluffy on addition of a little vinegar as explained here.