Does the school you are appealing for have an admissions category for social and medical need? If it does and the local education authority did not look at the medical reasons you had submitted - and they have already admitted this to your husband - then it sounds as if a mistake was made. However, the mistake is in not looking at the reasons, not necessarily in not giving you the place. That depends on whether you provided all the necessary supporting evidence. Usually, to get a place under the social/medical need category (where it exists, not every school has it), there has to be evidence from health care professionals (not just from the parents) that this school is the best one for meeting the child's needs. Did you provide evidence from a health care professional about your son's medical condition?
To be frank, what you have said puzzles me. You say that your child's medical problems are not significant and that he cannot walk far, but from what you say about being out of catchment, the school you are appealing for must be further from your home than the other nearby schools for which you did not apply. It might be (but I am speculating here) that even if the LEA had considered the medical reasons you had submitted, they would have taken the view that although your son has a medical issue, it isn't one that means he needs to attend this particular school.
You therefore need to get more information from the LEA aboit why they did not look at your medical reasons: was it because the school has no admissions category for social/medical need and so they could not give you priority on this basis? was it because you did not provide evidence to back up what you said about medical need? or was it just a mistake?
If there is an admissions category for social/medical need and you provided all the right evidence but the LEA just ignored it, then you might have grounds for saying that the LEA made a mistake which cost your son his place. In that case, you should win your appeal (and in fact the LEA should give you a place without making you go to appeal).