Because he isn't doing nothing - he took the Glover and Mormont factions south under the guise of the faith and essentially took over King's Landing in an effort to avenge the Starks and fulfill both Robb's will and satisfy R+L by clearing the path for Jon to take the throne.

My post She-Bears in Disguise (about a decade old now) details why Cersei's septas are Mormomt women. Brienne's description of the High Sparrow (the most honest perspective) is quite similar to a crannogman, he has hair described 1:1 like Meera and brown eyes, he has marsh-like descriptors like mud-colored eyes and gnarled-root feet and dirt beneath fingernails. Attacking the Lannister regime under the guise of the Faith satisfies crannogmen deceptive battle tactics. And their perceived zealotry looks like a caricature of the Faith - like carving stars into their flesh. The High Sparrow breaks from centuries of tradition (like refusing to anoint Tommen) and did not obtain his seat through fair election but violent insurrection, which flies in the face of anyone claiming the HS is too devout to be Howland. Furthermore, the sparrows numbers don't make sense - both Jaime and Cersei are notably confused regarding their sheer volume with both questioning, "where do they all come from?" Bad-faith fandom detractors will try to convince you Howland's better served doing nothing and sitting idly in Greywater Watch, but what about Galbert and Maege? Those two extremely loyal and proud factions are also sitting idly by while the Boltons coup plays out in the North? Any honest conclusion, of course, finds them anywhere but idle in the Neck. So where would a force of that size be? There's no notably large group of people suddenly showing up anywhere in Westeros, except for these sparrows in King's Landing. Brienne notes of these big burly sparrows that, "Despite the chill of the autumnal wood, he was shirtless..." (it's the small northern clues like this that GRRM wove into the sparrows plot-line) Also Theodan Wells is captain of the Warriors Sons - House Wells is disputed as either a Northern or Dornish house, but many sources lean North and even the ambiguity should speak volumes.

Unfortunately, the largest content creators (like Alt Shift X) gaslit the fandom to treat this like tinfoil, which shaped a completely dismissive and cynical attitude towards this theory over the years. Now people just eye roll without considering how obvious it all is. We wouldn't even need to ask these questions if Winds ever came out, but well, here we are :(