• Grammar school 11+ test results are adjusted for age by almost all councils (so being the eldest is a disadvantage)
No, it isn't. The point of the age adjustment (which is minor given the difference between an 11 or 12 year old is far less than say a 4 or 5 year old) is to equalise the results and remove disadvantage. A child who has had an additional year of education and experience will perform better - that is why they are age adjusted! So their adjusted result is likely to come out exactly the same, if not better because obviously the proven negative social, emotional and academic impacts of starting school have been removed by deferring.
• can stand out for being bigger/hitting puberty first
The opposite could equally be true for those starting younger. And I've heard far more tales of bullying or mental health issues on account of that, than being older. Do September born children - often just a few days older than deferred children - tend to get bullied on this basis? Sources?
• when you are older you have to explain to peers why you are in the wrong year
There is no "wrong" year. The effect of the decade old policy is that cohorts are now 16 months long and overlap, with those in the overlap having a choice of cohort. If anybody is too thick to grasp this or causes problems about it then the school needs to stamp that out like any bullying for whatever invented reason. If kids are going to bully they will always invent a reason. The answer is to deal with the bullies properly.
• If your kid has a problem later (parent seriously sick, they get sick etc) then you potentially could hold them back at that point rather than them falling 2 years out of step. You kind of have a year in reserve if you need it!
I've never heard of anybody doing this! I'm sure it may happen but it's hardly a major factor to consider. And would be highly dubious this would help a child in such situations anyway in most cases as it would simply add to the disruption. People can't plan their choices on the basis of catastrophic but highly improbable events. The only rational way to make a choice is to look at the data and statistical likelihood of outcomes, which is very clear on this matter, hence the option to defer being introduced.