See figure 1.3a - https://www.gov.uk/government/publications/migration-advisor...
Note that to the best of my knowledge, these numbers don't include the Afghan resettlement scheme which would further lower the proportion of employment driven visas.
Who indeed will pick the cotton.
Progressives (in the US at least) generally support immigration with protections and fair wages. They also recognize, rightfully, that systems built for decades upon exploitative practices (low wages, no protections) if removed overnight will cause mass disruption of those systems.
Neither of these is in any way supportive of slavery, modern or otherwise. The first - suggesting that immigrants be treated civilly and paid a living wage - has been fought tooth and nail by 'free market' literalists. The second - that there will be disruptions in social and economic systems when an entire workforce is suddenly removed from the systems that it has propped up for decades - is common sense and historically founded.
You're conflating these things to try to justify a talking point that was just created three months ago.