Although Swaziland -> Eswatini has gained ground so much that Wikipedia now uses the latter as the primary name.
Why? You don't call Istanbul Constantinople anymore.
Those are both different words from the previous, not less-familiar and less-phonetically-intuitive versions of the same word. Heck, in Eswatini's case, it's arguably easier to read and pronounce than Swaziland. Your parallels make no sense.
You think so? I think Swaziland is remarkably intuitive to read and pronounce. But thats probably because land is an English word to begin with.
The "land" part is fairly straightforward, sure, but "swazi" strikes me as a rather disharmonious combination of phonemes, and the two pieces don't mesh well together at all. "Eswatini" rolls off the tongue much more naturally to me. Maybe I'm thinking of it more from the perspective of a Romance language speaker than an English one though.
As someone who has English as a first language, 'Swaziland' is a lot more natural to pronounce than 'Eswatini'. The transition from -e to -sw is a pretty awkward one (no English words have these three letters at the start). Awkwardness of the pronunciation will vary considerably depending on the exact 'e' sound a language uses though.