As a side note in terms of fitness, in the US, the number eligible people is rather low due to the number of constrains the US military places on fitness, weight, tattoos, mental health etc. The number of people that could actually fight it everyone were called up that's within the age bracket is somewhere in the teens in terms of percentage. Unsure where that stands in Europe, but it would be a large roadbump if the draft were to be reimplemented.
They do, desperately so. There is a reason we have an entire internet economy of influencers and hucksters targeting in particular disoriented young men. They're pretty much begging for someone to provide meaning, discipline, comradery the kind of which traditionally you experience in for example military service and instead we leave them to Andrew Tate.
We have huge issues with young people delaying adulthood, not having structure, feeling useless and so on. Demanding someone serve their country is to take them seriously, give them responsibility and to provide them with a good reason to cultivate character.
Regarding self-selection and career prospects: We went to a military fair on a school trip in the 90s. There was an enlistment pitch, too, and even then (conscription had not yet been paused), they seemed to target those who didn't know what they wanted to do with their lives. (Everyone in my class was technically headed for tertiary education eligibility, so unemployment was a far-away concern.)
For one it instills in young people a sense of duty and direction and also simply physical fitness which many struggle with these days, in Germany an important aspect was always to have civilian supervision in the military ('citizen in uniform') not a sort of separated, private army. A lot of people in the army I talked to have noted that nowadays a lot of people who sign up are just people with limited job prospects. An army of the unemployed is ethically awful. National service should not be pushed on people who do it desperate for a paycheck. Everyone in the country benefits from its defense, so nobody should be able to skip out. I admire the culture of say Israel, Finland, Norway or Switzerland, small countries where this has always been on the mind of people.