That sounds very similar to my youngest. He is a very 'wrigglesome' child, and often appears not to be paying attention, but actually is. (And of course sometimes is not!). He is capable of concentrating, but struggles under certain circumstances.
To my mind it's very much a behaviour management issue, rather than a problem child. Some teachers cannot seem to cope with children who are not instantly compliant.
His best teacher so far was actually the least experienced one (can't remember the terminology, but it was her first year as a fully qualified teacher), closely followed by the most experienced one (who retired the year after she taught ds).
His behaviour reports were vastly better with those teachers, and he was clearly happy at school.
The teachers who could not cope with him were the one that shouted a lot and the one that did not engage with him at all. In those years he was unhappy at school.
This year's strategies have been:
His own packet of equipment, so that he doesn't have to go fetching whiteboard, ruler, number square, pencil, scissors etc from different locations in the classroom. Very successful, as he is able to settle himself into the work more easily.
A weighted pencil with finger grips. He is still struggling with writing. The pencil is helping, but I think we need to address this again.
Having him sit tucked well in to the table. Like the pencil, I think this is an anchoring technique that improved the physical feedback from his body. Sometimes this works, particularly after break or PE, but sometimes he needs to get the fidgets out, and it is counter-productive. In infants they would do a Wake-and-Shake for the whole class, and I think ds still needs that.
We're looking at the possibility of moving him around in the classroom, so that he ends up sitting at a solitary table for the focused writing parts of lessons. This is not a punishment, but a way of supporting him by moving him into his own zone. It is not something I would have agreed to with the teachers who I felt did not manage ds well, but this year's teacher is very engaged and I trust her attitude.