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Why are some absences of a limit bigger than other absences of a limit?

That's the advantage: they aren't. Though one quantity may "approach infinity" (diverge) faster.

Yeah I think this confuses rates of change with set cardinality.

“Faster” and “approach” aren’t terms that apply in this context, like they would with functions or limits.

It does seem to make sense to say that the natural numbers approach infinity faster than the (positive) odd numbers. Even twice as fast. I think that's why we say that half the naturals are odd.

It doesn't make sense at all, if you see them as a sequence (which is wrong, sets are not ordered) then after taking the first 3 elements the odd numbers are at 5 and the naturals are only at 3, clearly the odd numbers are approaching infinity faster.

If you don't see sets as a sequence then neither is approaching anything.

The natural numbers are ordered, yes, that's why it makes sense to talk about how "quickly" they grow. The unordered set theoretical notion of sizes of infinity arguably makes less sense here.

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