That is absolutely false. I know many people who have lived legally in the USA for many many years with valid visas and have intentionally never pursued a green card. Two people come to mind including one who has over 20 years the US on valid visas -- she intentionally never pursued the green card despite both (a) being married to an American and (b) being legally able to get the green card.
Some of them are now pursuing green cards only because of federal government's immigration enforcement not only going after illegal immigrants or criminals but clearly and intentionally pursing immigrants in general -- even those who are legal and without any criminal history.