I'm choosing from bad options here (not that it's my choice)
Yes, we should have purpose-built smoke tests. That's absolutely the preferred option. But that isn't what they built first. What we have today is the set of automated tests they're using in our test environment. Before the next deployment we can either put the test data in the production database -- in a separate table that can be deleted after the test -- or we don't run any automated tests after deployment.