Minimum Wage vs. EITC: Revisited

A while ago I posted this saying that the minimum wage and the EITC seem to be trying to solve the same problem, so we should end the less effective one (minimum wage) in favor of increasing the more effective one (EITC). But now I've come across a convincing reason to use a mix of the two here:

"The EITC partially subsidizes employers, and as such the minimum wage is an excellent way to combat this. So it complements, rather than substitutes, for an EITC."

Basically, if someone is willing to do a job for X dollars per hour, and then we start giving them a tax credit of Y dollars per hour, then they will now be willing to do that job for lower pay because of the benefit of the tax credit. So now the employer can pay the employee less (won't happen immediately, but that's how the market will adapt over time). In the most extreme case, their wages could be reduced to X-Y, which would lead to that tax credit, in effect, going entirely to the employer. So a minimum wage (by limiting how much wages can be reduced) lessens the amount that the EITC (intended to increase a worker's income) can get shifted to the employer.

This also has me wondering about how much the EITC can explain the rise in income inequality since the 70's. The EITC was started in 1975 and has been increased since then. Meanwhile, the minimum wage has mostly trended downward, adjusting for inflation, since 1970. Hmm...