You seem to be assuming that if the government doesn’t give these people the money, the employer will. Why is that?

Is it not also possible that the government benefits make low-wage work less appealing, thus lowering the supply, and increasing the compensation?

The lowest skilled workers at the bottom of the productivity scale don't generate enough value to justify paying them a living wage. If the company is forced to pay a living wage to such workers then management will send the work offshore, or automate it, or just not do it at all. The worker will then have zero wage.

Instead of artificially raising the minimum wage we should put more funding into job training. Help those marginal workers build the skills they need to earn higher wages.

No, giving money to employers is a subsidy for employers. Giving money to employees is a subsidy /against/ employers.