A Center for Immigration Studies’ examination of current USCIS datasets suggests that, of the nearly 13 million green card holders in the U.S., 5.5 million have expired green cards, and neither they nor the government seem to be worried about it. (It's important to note that an alien's permanent resident status doesn't lapse if the card expires.)
According to the most recent USCIS estimates, as of January of this year there were an estimated 12.88 million green card holders in the nation, 970,000 of them having arrived before 1980.
Since the vast majority of green cards are valid for 10 years, we might assume, given a fairly steady volume of legal immigration, that every year 1,288,000 of them would expire and would need replacement. This is a conservative estimate because under some circumstances a card would need to be replaced more than once every 10 years — some aliens change their names or report lost or stolen cards, while still others, for technical reasons, need new cards.
We compare the 1,288,000 to the number of applications for replacement cards that, over the last four years, have averaged 733,000 a year, a gap of 555,000; over a 10-year period this would add to 5.55 million aliens apparently with expired cards who have not filed for replacement ones.
And simply filing for the card does not produce it right away; as we noted earlier, as of September 30, 2021, the backlog for decisions on these applications was at the 625,000 level, a number that had risen by 280,000 over the prior 12 months.
So a slow-moving agency and a perhaps even slower-moving green card population, has led to the current situation.
Getting a green card is a major move for an alien; keeping it up to date, not so much.
For more detailed data on the flow of these applications, and the decisions (or non-decisions) that go with them, go the CIS data portal for USCIS under “All USCIS Application and Petition Form Types”. “Application to Replace Permanent Resident Card” is in the “Other” section of these documents.