I see. But not touching the minimum wage, and instead making the same amount of benefits provided gradually instead of having a sharp boundary (as now) would go a long way.

It won't make any businesses die; to the contrary, the people who won't take a slightly better job now would grow at work, and maybe eventually grow out of poverty entirely. Currently it's only possible by either taking a hit in income among the way, or by suddenly jumping to a much better job.