Our modern civilization can fall, too, and in a surprisingly short time. It's happened right before our eyes multiple times over the past half-century. Think of the Soviet Union in the late 1980's, or Venezuela over the past five years. It could happen even to the USA or Western Europe, if enough of the storm clouds gathering over either economy were to let loose. (Think of what would happen if the US government, due to economic circumstances, were to stop providing Social Security and/or Medicaid to its citizens: or, alternatively, if we did not act to stop illegal alien infiltration, and they did to the whole of the US economy what they've already done to California. What would either and/or both of those events do to a very large part of our population? And what would that imply for our society as a whole?)Maybe a light snack.
Food for thought.
Unlike Rome, the means of decisive intervention hereabouts is not solely vested in the government's minions.
Hordes of illegals doing to the US economy what they've done to CA would simply initiate a bloodbath reckoning, and expulsion via the local cemetery, and in haste.
Stopping Social Security or Medicaid would be equally harsh, but the old and infirm aren't liable to kick up much of a fuss as they died from starvation or lack of medical care in the same numbers as the prior scenario.
(And, I suspect, local charity would pick up much of the slack for the deserving poor, and the hopelessly addicted would die in droves or be re-institutionalized en masse, simultaneously ridding us of society's wandering gypsy refuse and central government's power, both salutary consummations devoutly to be wished.)
In many ways, society is both more fragile and more robust than it was in 400 A.D.
Wholesale breakdown is less likely, though certainly not impossible.
You've just chosen the two scenarios least likely to do more than inconvenience society for more than a season or three, for those other than the target demographic.
Neither A Day Without A Mexican nor A Day Without Grandpa would affect most of society except to lessen the traffic load on the interstates, and shorten the waiting time at the local ER, ultimately.
Bring down the financial system or the power grid rapidly and in total, and you've got quite a different matter on your hands.