I can see you have a lot of issues to think about!
It does sound like home ed would be best for your daughter academically for the time being. If she learns better at home than at school then obviously home ed is a better way to prepare her for entry to selective schools... though of course it might turn out that those selective schools don't suit her as well as home ed and you don't want to send her there in the end!
From your description, I wonder whether there may be some specific issue which makes a group setting tiring and unpleasant for her. That doesn't mean you have to move heaven and earth to keep her at school now and identify the problem ASAP - a more practical solution might be to take her out now and (in view of your plan to send her to school when she's older) explore the possible issues which might be causing difficulties in a group environment.
I don't know the list of things you should explore there but maybe if you post on a busier board someone can help. Hearing springs to my mind simply because my friend was discussing her daughter's hearing problems with me today. Her child can't hear well when there is background noise and has to work really hard to know what is going on, which leaves her exhausted. Or maybe she is distractible and needs a quiet peaceful environment? Or is sensitive to noises and lights, making a school classroom is so uncomfortable she can't concentrate there? Your daughter is very young. If you wait and observe her and experiment over the coming years, it may be possible to figure it out. Or when she is older she may be able to articulate the problem herself - though that is bound to be hard for her because how she feels is normal to her and she doesn't know that other people don't experience the world as she does. Whatever the problem, it's clear that school isn't working for her now.
I find this terrifying thought because just having her over the summer for a few weeks with no clubs resulted in her becoming a little clingy and not caring to see any other dc.
Home ed parents often report that it takes some months for a child to adjust to being away from school. Perhaps you could regard home ed as a temporary thing and give it a good long while - maybe a year? - to see how she will adapt.
Some kids get so used to the highly structured environment of school where they are told what to do all day that they literally don't know what to do with themselves when that structure disappears suddenly. It's a common complaint from parents of schoolchildren, that their kids are bored in the school holidays. The trouble is that though this boredom is likely to be temporary, it can take more than six weeks to work through it, meaning that kids who go to school never have the opportunity to do so.
Some kids don't like spending hours a day with other children, especially in a crowded place. They may find it tiring and stressful. If this is the case, when the requirement to do so is lifted they'll be so relieved that they want a while on their own to recover from that stress before they start craving the company of other kids (probably for fewer hours a day and in smaller groups than they had to do at school). So again, wanting to avoid other kids may well be a phase which needs time to get past, rather than a permanent condition.
Team sports - sounds like your dd's school does far more sport than a state primary would, and you see this as a bonus. It's difficult to replicate that wide range of daily sport elsewhere; my area is unusual in having PE-type home ed sports lessons but it's only once a week.
On the other hand if you are talking about a few specific sports then depending on the sport, the out-of-school opportunities could be at least as good as at school, plus you can choose the specific club to suit your daughter best. For example, if she likes football then you could choose from several different local football teams. This will take up more of your time, however, as you'll need to ferry your daughter around rather than dropping her off at school and having it all provided there.
High-level competitive sport is actually far easier for home educated kids to do. I know many HE kids who do this; in fact some parents take their kids out of school for this very reason. As you've seen, home ed is much more efficient than school, meaning you only need to spend a few hours a day at most on academics. That frees up time for sport, so the child needn't be chronically busy and exhausted. And a home ed schedule can be totally flexible to accommodate sports training and events.