Pretty sure the house and senate office buildings also both contain these.
However, I know of at least one other building in this area, the Hobart Building, slightly younger but much taller, where the iron staircases are still open and still serve their aesthetic function. It makes taking the stairs so much more pleasant.
Apparently the fire risk is manageable. I think a bigger problem is that these building were designed for a multitude of small office suites. But today most of these buildings will have (or want to have) tenants who wish to lease entire floors, or most of a floor. Large, open, but single tenant floor spaces are how most modern office towers are constructed, so that's the market expectation.[1] It's difficult to provide such tenant spaces while also keeping open corridors between floors. Even for mail chutes, as the private, controlled-access floors diminish the desirability and utility of shared facilities.
[1] The nearby Crown Zellerbach Building (1959, https://en.wikipedia.org/wiki/One_Bush_Plaza) was at the vanguard of this office tower design evolution.
The sixth (1970) and seventh (1981) editions of Architectural Graphic Standards discuss them on the “Planning for Postal Service in Office Buildings” pages. From the latter:
CHUTES: Used in buildings of at least four stories.
The chute must be approximately 2 x 8 in. in cross
section and extend in a continuously vertical line
from the beginning point to the receiving box or
mailroom. The interior of the chute must be access-
ible throughout its entire length. Chutes installed
in pairs are constructed with a divider and dual
receiving boxes. Chutes are for first class mail only.