I believe it will effect prices far more than you think it would, or likely should, because all the middlemen between farm and the person actually eating it are not going to just accept lower margins themselves if prices rise, and they some might even demand higher margins on food because they know it is in short supply and there won't be enough available for new competitors to jump in and try to undercut them. The redistributor, the processor, the packager, and the retailer side are not going to accept flat profits with lower margin percentages when they have a supply during an overall shortage of those same goods.
If farm laborer wages just went up and nothing else changed, it would probably be fine. If wages could stay the same but there was a small decrease in productivity from less experienced workers filling in, it would probably be fine too. But if the lack of workers is what is causing both production shortages and sharp labor cost increases at the same time, the only people with any power to wield in this situation are the large corporate middlemen who hold tons of supply contracts with legally binding guarantees for product and an entire logistical network developed over decades to transport and sell these goods that can't just be spun up in a few weeks by other people.
If farm laborer wages just went up and nothing else changed, it would probably be fine. If wages could stay the same but there was a small decrease in productivity from less experienced workers filling in, it would probably be fine too. But if the lack of workers is what is causing both production shortages and sharp labor cost increases at the same time, the only people with any power to wield in this situation are the large corporate middlemen who hold tons of supply contracts with legally binding guarantees for product and an entire logistical network developed over decades to transport and sell these goods that can't just be spun up in a few weeks by other people.