Lawyer here.

The short answer is: enormous expansion of federal and local government regulation and ability to enforce those regulations.

The long answer is:

The EPA was created in 1970. At first, it was a relatively small government body. However, in the 1990s, the Clean Air Act Amendments were passed into law. These amendments were passed with "overwhelming bipartisan support." www.epa.gov clean air act overview. While the Clean Air Act didn't directly ban "barrel fires," it did provide a framework for regulation of air quality pollutants and reflected the ongoing shift in the culture of the 1990s "Race to Save the Planet."

Why does this impact you in Washington State? Well, as I mentioned above, as local governments expanded along with the increased regulation, curbing all "outdoor burning" was seen as a low-resistance way to match the federal EPA clean air effort. From the State of Washington's Department of Ecology: "Smoke from outdoor burning pollutes the air, causing serious health problems. Much like cigarette smoke, the smoke from burning leaves, grass, brush, and tree needles can cause asthma, emphysema, bronchitis, and lung cancer. Children, the elderly, and those with breathing problems are harmed the most by poor air quality." https://ecology.wa.gov/

Note how closely that tracks the 1990 EPA Clean Air Act highlights: "Air quality has improved significantly in our nation’s cities and towns, reducing health threats such as lung damage, asthma, heart attacks and premature death." https://www.epa.gov

Specifically, in 2000, the State of Washington banned barrel fires entirely. Again, the reasoning was under the umbrella of health, safety and welfare regulation; the same rational that supported the Clean Air Act.

To directly answer your question, you don't see barrel burning anymore because people/governments wanted to clean up the air/cities. Barrel burning was identified as a source of air pollution, and it was regulated out of the society over a period of years.