Jabed,
I had a very bright 6 year old who i withdrew from his village school after a year and a half because it was damaging him. I would not be so rude about it as you are about yours - it served him very well in Reception with a teacher who was interested in him and very prepared to work with his 'differences'. However, in Year 1 the combination of a weak teacher, a mixed class with some very difficult children in it, and an inflexible attitude towards DS doing different work from the others created a significant problem. DS became very anxious, a selective mute, and showed very significant ASD traits.
One of the solutions that the school proposed for him was to move him, as a Year 1, into a year 3/4 class - so to accelerate him by at least 2 years. I rejected that, and home educated hium until we - coincidentally but very luckily - had to move area because of a change in my husband's job.
As the root of DS's difficulty in school was predominantly social (good teachers in the state sector since have remained perfectly able to challenge him academically), I was determined that he should have a good mix of 'socialisation' within HE. We tried the local Home Ed group - do you have one? - and although DS found the chaos of full meetings impossible to handle at the time, we went on lots of trips with them and he really benefitted from that 'doing something interesting with a group of other children of mixed ages' thing. We also met up on a one to one basis with a couple of local families with HE children around DS's age, who we wouldn't have met had we not attended the group.
He was already a Beaver, and I have to say (he is still a Scout) that the Scouting movement has been fantastic for him with its mix of active but not sporting activities and well-controlled socialisation within a clear framework. Woodcraft Folk is an alternative for those who find the Christian ethos which theoretically (mostly invisibly) underpins Scouting too much. Google Beavers and the website will show your local contact. There may not be a group in your village, but you are likely to find groups easily within a short journey. Groups differ - if you have several to choose from, it might be a good idea to visit a few as the quality and variety of activities and the purposefulness of the meetings does vary with the leadership.
The village happened to have a football club, which DS joined. Not because he loved football - though in fact he turned out to be a very, very good goalkeeper who has since been scouted by our local professional team - but because 'being in a team' was something that he needed to learn alongside maths and science to be properly educated.
DS also had weekly swimming lessons in a group - that 'doing something alongside others' where an adult's attention is not for one child alone was also a good antidote to HE. For this - as for rugby, which he also did for a while with his dad, as I did most of the HE and it was important that they did things together too - we travelled a few miles to the nearest town. One of the things about HEing in a village is that there are limited activities available locally, so you do have to be prepared to travel to access some of the opportunities that would benefit your child. I have described what DS did - but had he been interested in different things we might have travelled for group music-making, for gymnastics classes, for science clubs, for ice-skating - for opportunities to persue those interests in a group setting 9which is important) we would have had to travel.
In the holidays - may not be possible for you as you are abroad so much - there are lots and lots of 'activity' weeks, mainly run for holiday childcare for schooled children. I found them really good for DS while HEing as they were very organised, often had very high quality input (DS did a football one run by the local professional club, for example) and gave him whole days away from me and under the care of different adults.
In the end, our long term solution was to move, and return DS to a different primary in our new town, where he has absolutely thrived both academically and socially. However, had we had to continue HE then I think a balance of Beavers, local groups, a link with other HEing families with similar philosophies (We were 'structured' HEers, and had several similar families within a radius of 5 miles or so) and travelling to special interest groups away from the village would have given him a reasonable balance.