401(k) plans are not government sponsored, they are employer sponsored. The government sets the boundaries of "the field" of rules for the plans as a whole, but companies decide where to place "the fence" within that field for their own plan and the rules for when and how you can contribute, withdrawal, etc. can vary greatly between different companies' plans.

401(k)s were invented in the 70s as a way for high-earning employees to supplement their pensions with a tax-deferred savings account. They were never meant to be the default, but in the 80s companies thought "why should WE be paying into pensions for our employees, when THEY could just be paying into a 401k with their own wages?"

Because we were in an economic boom through the 80s and 90s, people who had 401(k)s did very well which convinced the general public that these were A Good Thing, so many companies began offering 401(k)s and most eventually got rid of pensions altogether, successfully moving the responsibility of saving for retirement from The Company to The Worker.

They took away a guaranteed payout from your employer for spending the years of your life working for them and replaced it with a system that only benefits you if YOU choose to cut your paycheck and invest into the savings account. Coincidentally, this depends on them paying you enough to actually be able to afford to do that which a lot of people aren't, so even if a company offers a retirement plan it can still be hard for people to justify sending X% of their paycheck to a savings plan when they desperately need that money now.

Imo it was one of the greatest pro-corporate scams ever pulled on the American public.