Once their immigration status is tied to their employer they have no ability to shop around for better pay.

So why would they enter the program? They currently have demonstrated that they have no problem not having an immigration status, so why would they switch to having something that doesn't benefit them, that they don't want, and that costs them money?

Their goal is to make the legal path cheaper to appeal to farmers, but farmers aren't the ones driving the price. As you said: market rate is higher than this guarantees people. If there's a growing shortage of labor you can expect labor wages to rise. Why would you agree to work for less if you can just go to a different farm and make more?

I understand your point and the situation perfectly well.

migrant labor would get more and more expensive, as more and more people are deported.

I believe this is why you're wrong, and farmers aren't hoping it goes faster, but rather voted again their own interests like so many have, and just didn't think they would specifically target their livelihoods.

A racist administration deporting people aggressively, lowering the incentives to come here legally, and not caring about the consequences, while farmers scramble to control damage they didn't think was actually going to happen is a way simpler story. Also fits nicely with "America first" burning the ability of those farmers to sell to a global market, canceling programs that gave them money, and canceling food aid orders that mostly existed as back handed subsidies.