This comment does a good job of breaking down the origins of the conspiracy theory and the (many) times its claims have been investigated and refuted, but I have not been able to find out why the flags are everywhere.
I wrote the original comment you linked to, but I didn't talk much about the flag in that post. The POW flag was created by the National League of Families in 1972, and though we should note that in 1972 there were indeed actual American POWs still being held by North Vietnam, post-1973 the League has continued to be a leading promoter of the false conspiracy theory that POWs were left behind after the war.
Ronald Reagan was the first President to fly the POW flag over the White House in 1982. Others could write more about how Reagan's political campaign utilized the public memory of the Vietnam War in an era of post-Vietnam disillusionment with government. There was a sense (which someone else has probably posted about in more detail about the extent to which this was true or false) that the government had let Vietnam veterans down by not providing them with adequate support & resources (many returned home with drug addictions, health problems as a result of the war, or PTSD) and that veterans had been unfairly scorned by anti-war protestors. Part of Reagan's appeal to the public in the post-Vietnam period was that he, as Maureen Dowd wrote in the New York Times in 1990 "made us feel good about ourselves and our country at a time when we were tired of being called second-rate" In that political climate, showing support for veterans and former POWs was never going to be a controversial position for Reagan to take at that time, even if people didn't dig too deeply into what the conspiracy claimed.
In 1990, Congress passed Public Law 101-355, recognizing the efforts of the National League of Families, giving the POW flag official status, and stating, for the record in an official act of Congress that there were “Americans still prisoner” in Southeast Asia (17 years after the real last POW had returned home). The flag became common on military bases and became part of military culture. US service members and ROTC cadets frequently hold “Warrior Runs”, where participants run for 24 hours continuously carrying the POW flag. The POW myth has percolated into military culture, but the flag is often seen as a non-conspiratorial symbol of support for veterans in general.
The display of the flag on government buildings has not diminished as the memory of the Vietnam War fades. Public Law 116-67, passed in 2019 (sponsored by Sen. Elizabeth Warren), now requires the POW flag to be displayed every day the US flag is flying over a government building. The flag is now inescapable at all federal installations and it has official status. It is hard to imagine a flag created by another activist group, such as Black Lives Matter, on mandatory display at every federal government building, but as something that on the surface looks like a non-controversial sign of support for veterans, it has widely spread in the subsequent years since the last Vietnam POW was released.
References *Public law 101-355 Designating September 21, 1990, as "National POW/MIA Recognition Day", and recognizing the National League of Families POW/MIA flag: https://www.govinfo.gov/content/pkg/STATUTE-104/pdf/STATUTE-104-Pg416.pdf#page=1 *Public Law 116 - 67 - National POW/MIA Flag Act https://www.govinfo.gov/content/pkg/BILLS-116s693enr/pdf/BILLS-116s693enr.pdf