Yes, I really think people imagine what home education would be like and this leads to unfounded speculation. If you want to know the effect that home education can have on anxious children, the people to ask are parents who have actually done it.
I have known many families where children came out of school due to anxiety. There are various causes for the anxiety. Some kids quite rationally dislike school because it is such a poor fit for them. Perhaps they find the noise and crowding overwhelming, which might indicate they have SEN, but might just be their character. Perhaps they are still not mature enough to choose to be away for long stretches from loving adults who have time for them - that too is within the normal range and does not indicate any pathology. Perhaps there are actual issues with the school such as low-level bullying, but often children aren't able to articulate this while they are in the situation because they accept that that is how school is.
Some children do have genuine separation anxiety. That doesn't mean that forcing frequent prolonged separation at an early age is the correct approach, or that they cannot get better and learn to deal with it if removed from school. Suffering teaches resignation, but it doesn't always teach resilience.
If you have your children with you much of the time, they may feel more able to separate gradually in a way which feels right to them as individuals. My eldest was an outgoing child who went on long playdates without me from the age of three, and roamed our medium-sized city alone on the bus from eight. My younger child was seldom over 20 metres away from me until she was over eight, and didn't really do solo playdates until she was 11. She's now a friendly confident teenager. She didn't need to be pushed toward independence against her will. She just needed the security of my presence for longer.
Home education doesn't have to be forever. You can try it for a year and see how you and your children get on with it. Many home educating parents do work, though usually at least one parent is only part-time. Perhaps you can find a warm childminder who can offer them a pleasant home-like environment, which they might adapt to better than school. My eldest loved going to a CM while I worked part-time. They played with the CM's kids (also home educated), went out to home ed activities, and chilled at her home.