At least part of the motivation for talking about contribution and insurance is the issue of migration, and EU migration in particular. If EU citizens can move to another EU state whenever they like, and be subject to the same rules as those who are already there, then there's nothing to stop people being in one place to take advantage of low taxes, another for high welfare, and another for better health care and so on. For fairness to truly exist in this would require either a uniform system across the EU, or a de-territorialisation such that each person would remain a member of a particular system wherever they were living. This latter option would mean thinking about what citizenship, in the broad sense of the word (taxes, policing, voting, benefits, passport, membership), would cover.
What movement in space draws attention to, however, is a problem of time. Much of the debate on migrants and the NHS and benefits is focused on how long a person should be in the UK before they are eligible for healthcare and the dole. Should it be the day they arrive, 2 years, 5 years, on getting a passport, or never? And what do we do about people who arrive from elsewhere but have a British passport? Should 21 year olds who've always lived here get benefits before they've done a day's work?
What I would point out is that comparisons to insurance should be reminding us of two things. First, insurance does imply people contributing and getting nothing out, and others contributing less and getting lots out: insurance is supposed to iron out luck and bad luck. Second, insurance starts from day 1: it's possible to buy a car, insure it, and crash it on the same day, and then get the pay out.
We can't predict how someone will fare in the future. A teenager who claims benefits might then get a job and pay far more in. Someone who works for a few years from 18-22 may well then never work again. Many people pay in for a state pension, and then die before they get old enough to claim it. Similarly, the migrant who arrives and claims benefits might then get a job and pay in for decades, and another might pay in for a few years, then claim benefits for many years. For justice, what we are really interested in is stopping people playing the system. This would require a much better idea of people's trajectories, motivations, and so on, but this is probably impossible. It's also part of some of the longer term approaches to these issues: even back in 1996 I was subject to a 'habitual residence test', such that a few months out of the UK made me ineligible for UK benefits. These kind of predictions are just too difficult to do, so instead politicians of all stripes resort to simplistic categorisations, whether justice is served or not.