As other posters have noted, one of the biggest sources of lead is window trim. Replacing windows in old buildings is also a serious renovation—for example, I know a landlord who paid about $15,000 to replace the windows in a 3000-sq. ft. duplex about 20 years ago. And that doesn't fix door trim, baseboards, or flaking walls.
But more importantly, I've never met a landlord who'd let a tenant replace windows or sheetrock. And what tenant would spend $10,000+ renovating an apartment they don't own? Certainly this is not a viable solution for government-managed, low-income rental housing.
A large landlord, on the other hand, can amortize the cost over multiple buildings and multiple years, and they can order correctly-fitted replacement windows in bulk. It makes no financial sense for low-income tenants to do expensive, piecemeal lead remediation. It's just another long-term maintenance cost, like replacing the linoleum floors, fixing the roof, or replacing the building furnace. (Of course, it sounds like the housing authority wasn't doing that reliably, either.)