Before revision:  More specifically, we show that the Myerson regularity condition plays no role in this constrained problem of welfare approximation. In addition to welfare approximation, we also study profit approximation using similar mechanisms but with a different threshold and show that a fraction of the optimal profit can be achieved with a large number of participants.
After revision:  In contrast, previous mechanisms proposed to solve this question, such as the second-best mechanism, typically require the knowledge of the entire valuation distribution. Also, our study extends the results to non-binary decision environments with general utility functions. Lastly, we show that if the threshold is instead set equal to the cost, the proposed mechanism can achieve a non-negligible fraction of the optimal profit can be achieved with a large number of participants.