The phrase "to thicken the pudding" means to make something more convincing or substantial. It refers to adding proof or evidence to support a claim or strengthen an argument. Full definition
I don't know if tapioca starch would work to thicken pudding without making it gummy, as it is a rather stretchy starch. (glutenfreeonashoestring.com)
Adding chia seeds thickens the pudding (which is almost mousse - like), and while they do break down in the food processor, their presences prevents things from becoming 100 % silky. (domesticate-me.com)