The question is - if you have certain privileges available only to citizens, how should you go about ensuring that only citizens are exercising those privileges? Barring some kind of identification registry - how would you do it?
Well, there's these things called living addresses, birth certificates, and tax forms. Along those lines. Stuff that's either pretty hard to get and maintain without being a citizen, or means they're paying anyway (i.e. acting like a citizen de facto, if not de jure) and about the only reason to really care is ideological
(and, just sayin', but ideological purity only goes so far towards keeping our roads from falling apart even faster). Makes figuring out if the person's a citizen pretty easy, most of the time. Some folks will probably slip through, but here's a secret when it comes to wide scale administration: If not enough people are getting through to break the system as a whole, it functionally doesn't matter if some
are, particularly when that subset isn't enough to meaningfully strain the system in question and/or costs more to fix than it does to let ride.* The optimization point is
between full enforcement and no enforcement, not one or the other.
Beyond that, though, the point isn't actually to make sure it's all and only citizens. Mostly is good enough, and mostly is much easier, far less likely to have to shit all over those mentioned privileges to obtain, and costs bucketloads less t'boot. Indicates your country has a moral character worth two damns, too, since you'll treat even folks that aren't one of yours well enough when they're on your land.**
As for the second question, if indeed the country was failing to ensure citizens had access to any and all privileges thereto, there wouldn't be much point. There also wouldn't be a country, for what that's worth. Still, there could be advantages to being a citizen even if you don't have much in the way of official government support -- various business concerns, support from fellow citizens (brought about by human psychology if not legal enforcement), all those sorts of things. There's more gains from being a citizen than just what the government on top of it dole out, basically.
That said, falling short of absolute exclusionary perfection on a few rights generally still manages to leave an pretty massive amount of advantages for citizens, in most cases, and is usually pretty excusable/ignorable/understandable when managing that perfection entails a logistical nightmare just as a starting point. The equivalent of a few percent's worth of your population getting limited access to some of those isn't exactly an apocalypse scenario.
* The latter is why a fair amount of welfare et al has a lot of pressure to abandon or avoid means testing, by the by. Active enforcement has this nasty habit of costing more than just kinda'... not.
** Tip for the unknowing: When it comes to legal et al matters, pretty much
every constitutional and most legal protections applies to non-citizens as well when they're in US jurisdiction. Closest to an exception you can get there is repatriation (to the originating country) or contested jurisdiction (and
usually no one wants that, so folks figure out which court someone's going to be tried in pretty quick). It's one of the reasons being an undocumented immigrant (and a good chunk of the stuff surrounding that)
isn't a criminal offense -- if it was, they would be entitled to the full protection of american due process and all the rigmarole surrounding it.
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... all that said, most of the ire you're going to see is less about the issues relating to citizenship and more about the ones related to the problems involved with enforcing your mentioned attempts. Conceptually it's a good thing and most folks'll get in on it to some degree or another (there's a couple exceptions loitering around GD, but complete or near complete abolishment of citizenship and whatnot isn't a common stance), but when it
practically means the harassment and abuse of people, citizens and otherwise, the erosion of rule of law and moral character of your country (this is particularly an issue for the US, as the thing was
built on immigrants, most of which were not exactly well documented), a general reduction in various rights, and a number of other things of ill extent (
especially including excessive cost, when the funds and effort could instead be going to stuff like making sure people have running water, stable power grids, and functioning roads), yeah, people are going to start getting pretty concerned.