Governments need to make sure that people can access the services that they’re entitled to through a wide variety of channels, including physically visiting an office if necessary.
Though I will say, at a practical level, you will find that it’s increasingly difficult for people with criminal records to travel internationally (due to entry requirements).
"Eligible" means you're able to be elected, but you must still be elected. Different from a passport, you may be eligible to a visa, and at some point an officer is likely going to interview you and decide whether to give you one.
People are entitled to a passport but only eligible to a visa. You can assert what you're entitled to, but not what you're eligible to.
> entitlement implies to me that the person has invested some time and resources in asserting their right
That's only when your title was earned, which not all are; some are born into them.
> I believe eligible is defined as meeting some criteria for, not necessarily being chosen though that could be a criteria.
The confusion may have started when decisions became more automated into "criteria" to be checked for by bureaucrats that no longer have the deciding power they once had (and later further automated by software), but "chosen" is in the latin root of the word. For example, "chosen" in Spanish is "elegido", "choosable" would be "elegible". "Eligible" = "electable" = "choosable". They're all basically synonyms.
Something else about what you said here, for titles that are earned (e.g. naturalized citizens), you don't invest time and resources to assert your right. You invest time and resources to earn the title. When you've earned the title, you've earned the rights that come with it. Having then those rights, you can then assert them. You don't need to expend anything to assert. You just claim them, since they're already your own. For example, if someone says you need to expend time and resources to assert your right to vote when you're already a citizen, that's wrong. Having expended time and resources to become a citizen (or having been born a citizen), it's already your right to vote. You're entitled to a vote.