No, it's broken because we've allowed millions of foreigners to come in and raise said healthcare and housing costs. Checking that people here actually belong here else they're deported is part of the "cost of living" solution, in addition to crime.
The idea that deporting every undocumented person will reduce the bloat and profit extraction in our health care system is making me giggle.
And do you really think checking every person is a strategy that scales? Why not just indiscriminately jail the people who hire them and thereby create a strong incentive to come?
Maybe those workers actually belong here more than you'd like to admit, but the powers that be enjoy keeping their status uncertain to use as a piñata they can beat whenever they need political candy.