Passports Were a “Temporary” War Measure
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I think it is possible actually, but the world would be much less "sharing", and social care would be developed much less (like in USA some time ago, where you basically had to survive all by yourself).
As a russian, I've encountered this few times, that countries are not very happy to invite me – I have to apply for UK visa ~6 months in advance with full itinerary, and same for few other countries with strict rules. Other are easier, but still, for the "developed" world, I am at the level of some refugee. Yeah, I know, I can have 5 previous successfull visas, have few millions saved in my homeland, but as a young person, it is pretty hard – and is true discrimanation; but I understand it's roots completely and don't complain about it, it just makes me laugh how different one's possibilities could be based on random things.
Most of the world by population and by land area.
That wouldn't be pleasant for either side.
Presumably, if one demonstrates mastery of language(1) and cultural knowledge(2), he can be assumed to be a countryman.
1) This practice dates back to biblical times. "Shibboleth": mid 17th cent.: from Hebrew šibbōleṯ ‘ear of corn,’ used as a test of nationality by its difficult pronunciation (Judg. 12:6).
2) Common example: During WW2, if a suspected spy were encountered, mastery of language and accent may not be enough to assess his status. Cultural questions were asked: Where does the NY Mets play? What's Mickey Mouse's girlfriend's name? Questions could also be about celebrities, movie stars, music, novels, etc.
In a case of genuine open borders criminals can leave the state quickly to avoid punishment. Then move on to a new target country.
This doesn't happen only because no country signs an open border agreement without also setting up a legal framework to punish criminals who would try to exploit borders to escape punishment.
The reality is that there are bad actors out there and if you aren't stopping them at borders then you'll have to start watching out for them closer to home.
High trust nations are like unpatched machines. You need a firewall.
Schengen has a shared outer border.
The development of the passport was more or less inevitable. Internal state-issued identification is a necessity at least for the right to vote if not also many domestic services from law enforcement (show us ID so we can ensure there isn't a warrant out for your arrest) to healthcare (show us ID so we can look up your health history, allergies, etc). The passport is merely a very-limited global standard for such identification.
We, as individuals, are not judged as such but rather as members of the "tribe" we accidentally belong to, whether we agree with the leadership or not or have any part whatsoever in their conduct.
As Spock would say, highly illogical.
Not sure how a fairer system would work though.