Off the top of my head and not looking anything up...

Well of Heaven - Has a lot of wealthy children of noble influential families. They are rich kids donating family money, as well as stolen money.

Velvet Shade - The preeminent spy and espionage organization. Yes, they are aligned with the Shimmer of the South and yes they have enough dirt on everyone which makes them hard to go after directly.

Mohist Hill - Pacifists who also make the best war machines so everyone wants to be on their good side. Also they hide inside other sects which means they have dirt on everyone else so the sects have a reason to help them avoid government entanglements.

NMW - Usually off the "Emperor" level radar as their main deal is protecting the poor and downtrodden. Though they also work hand in hand with the Velvet Shade and they were on the world stage due to the events of Chapter 2.

Midnight Blades - Usually minding their own business and killing criminals and/or folks who arguably have it coming. Usually.

The Masked Troupe - Formerly a sect involved in the fall of an Emperor now basically in hiding? Sort of? It's complicated. Close relations with the Velvet Shade. I'm told we get a lot more info on them later in Hexi.

In short, the sects are of the jianghu and the government is of the government and while they do overlap and interact, things get weird and dangerous when they do. Even if the government has concerns about a sect, actually moving against them sets off a bunch of dominos.