Ignoring the fact that the ash COSTS the mob money in lost productivity in their toxic castles, simple weather should have turned the area into a lush forest decades ago. You have to work REALLY hard and throw lots of money away to stop Mother Nature from springing back.
When Mt. St. Helens blew (1980), plants started appearing after 3 years. By 2000 the fauna had almost reached pre-eruption levels. Now that's not the same as 200 year old first growth timber, but it is very habitable.
Not thinking about it is fine, so long as you don't plan to do anything about it. But what if I decide I want to set up base in the Barrens? What if I want to plant some awakened trees for medicinal value? Is the ground fertile? Are there spirits actively killing stuff? Is there some whack job out there throwing dust in the air?
Don't forget, they've always laid it on pretty thick with the acid rain and other terrible pollutants, as keystones of their dystopic future and excuses for why nature shrivels up and dies all over the place. Full on "lush forest" doesn't happen very often in the Sixth World, unless someone is going out of their way to
make it happen (often magically, like with the Tir or some of the zaniness in the rain forests). The Puyallup Barrens suck because the books
say they've always sucked. Work out details with your GM, campaign to campaign.
I mean, I know that "because" isn't the most helpful answer, but it's really the only answer we've
got sometimes. I'm not just trying to be difficult, or dismiss your concerns, or whatever. It's just that realism hasn't ever exactly been the ultimate goal of the Shadowrun line (especially early on, when many of these precedents were set), and with 20+ years of momentum built on "it is ________ because we say it is __________," it's a little late in the game for us to try and rationalize or explain too much, without some real backpedaling and retcons.
