IMO, the short-term profitability of the company is irrelevant to the market rate they pay their workers.
If they pay too little, they don't get enough workers. If they pay enough, they do. Long-term profitability / cash flow / available capital/debt all affect the corporations ability to continue to pay wages, so in that sense they set a ceiling.
In terms of the government assistance, that is largely unrelated to the employer(s). There is a government program that determines person X is eligible for assistance and person X takes advantage of that program. It is not required that corporations pay an amount such that person X becomes ineligible for the program. For all I know, the eligibility rules take into account family size, student loans, health situation, alimony, or any number of other factors that are literally not the business nor concern of the employer.
An employer hires a worker to do a job. They pay that worker, after which point, the sides are square. The fact that a worker is or is not eligible for that program represents a subsidy to that citizen. If the company fired them or they quit, they'd still be eligible for that subsidy (in most cases). It's the decision of society to support its members, and I don't accept the translation to "it's a subsidy of the employer" as being a lossless one.
Enacting rules to ensure that companies need to pay more for employees who are on/near welfare or other government programs will likely have a negative effect on hiring those very workers, most of whom are presumably in a fairly precarious situation economically.
Do you want Walmart hiring Biff and Muffy, who are not on assistance because of their family situation, rather than hiring Joe and Mary, who are each single parents trying to earn money to lead their kids out of poverty?