Didn't Bran state somewhere that every tree with a face is a greenseer? Also don't almost all weirwood trees have faces on them?

Yes, almost all weirwood trees have faces, but they are just a camera, and do nothing without a greenseer looking through them. Greenseers can hook themselves up to trees, like Bloodraven and other greenseers in that cave did, but that does not mean that each tree is a greenseer. They're not sentient.

I doubt Harrengall curse is caused by the weirwoods. If that was the case, the Small Council would be cursed.

And weirwood isn't just used in important places of power. Some was used for the doors of a smith shop in King's Landing:

Ned turned off the square where the Street of Steel began and followed its winding path up a long hill, past blacksmiths working at open forges, freeriders haggling over mail shirts, and grizzled ironmongers selling old blades and razors from their wagons. The farther they climbed, the larger the buildings grew. The man they wanted was all the way at the top of the hill, in a huge house of timber and plaster whose upper stories loomed over the narrow street. The double doors showed a hunting scene carved in ebony and weirwood. A pair of stone knights stood sentry at the entrance, armored in fanciful suits of polished red steel that transformed them into griffin and unicorn. Ned left his horse with Jacks and shouldered his way inside.

The moon door, while a memorable thing, isn't that important, and it is made of weirwood:

The press of spectators parted. A narrow weirwood door stood between two slender marble pillars, a crescent moon carved in the white wood. Those standing closest edged backward as a pair of guardsmen marched through. One man removed the heavy bronze bars; the second pulled the door inward. Their blue cloaks rose snapping from their shoulders, caught in the sudden gust of wind that came howling through the open door. Beyond was the emptiness of the night sky, speckled with cold uncaring stars.