We had that to some extent in the pandemic.
The rich and those who saw themselves as the elite opted out of that social experience. It was seen as a money making exercise.
Any future iteration of national service will be seen as something for 'them' not 'us' .
Mid 20th century National Service certainly taught a section of society how firearms and explosives could be used, and how to form & maintain gangs through hierarchy.
The formation of the NHS wasn't as egalitarian as we think, with opt-outs for private medical practice, the continuation of private hospitals & private wards, and the scandalous GP contract that we are still suffering with.
I agree with your premise that we are a divided society but I'm not sure I see your solution.