Why does Randyll Tarly mete out a far more severe punishment to a thief for “stealing from the seven” one moment, and then disparage the authority of the Seven’s emissary, the High Sparrow, as the “twitterings of sparrows” the next? Is he merely towing the Tyrell line?

At bedrock, Randyll Tarly is a fundamentalist when it comes to the very specific worldview of the military aristocrat, and everything else subordinates to that. Hence why he would brutally execute a kinslayer, but still threatens to murder his own son because he thinks he’ll be a weak lord. 

So Tarly is definitely a follower of the Seven – the Smith is at the bottom, the Warrior on top of him, in obedience to the Father, and the rest handle women’s matters and death – but he’s not particularly pious, because that’s what septons and women are supposed to do while he’s busy chopping people’s heads off. 

When he looks at the High Sparrow, he sees an up-jumped commoner who took the title by force from men from the right class, who’s overturned hundreds of years of tradition and royal authority in assuming powers that rightfully belong only the King, and who’s encouraging rebellion and the disruption of the feudal system. So he’d happily cut the man’s head off, make the next High Sparrow absolve him of the sin, wash his hands, and consider himself entirely in the right.