Do people continue to use "il est" in this way? Or has this sense of the phrase faded away?

While you shouldn’t expect to hear it in casual conversation, the sense of this phrase has not faded away. It still appears either in very formal speaking or in written French. e.g.:

Il n’est pas de jour où je ne me connecte sur Internet.

Il est des choses qu’une mère ne peut raconter à son fils. Olivier Deck, Le chant des passereaux, 2012

Diacre, il est des choses qui ne sont pas de ton ressort. N’y aventure pas ton grand nez. Cavanna, Le sang de clovis, 2001

Il est des jours où on voudrait avoir eu tort d’avoir prédit, avec beaucoup d’autres, que les désordres écologiques, financiers, politiques et culturels finiraient par ne plus être des hypothèses Jacques Attali, Perspectives économiques, 2012

Je me souviens que quelques jours après cette déclaration en forme d’aveu, la Société des Compositeurs et Auteurs déclarait : « (…) s’il est des excuses à présenter, je crois que c’est plutôt TF1 qui devrait les présenter à la France. » plumedepresse.net

While very close, there is a slight difference between il est and il y a. The former must introduce something generic while the latter can also introduce something specific, e.g.:

Il est des gens qui se contentent de peu. Generic; a self sufficient, indefinite and impersonal statement. (People who content themselves with not a lot exist)

Il y a des gens qui se contentent de peu. Specific; personal; some examples might follow. (There are people who are contenting themselves with not a lot).

That means il est cannot always replace il y a. Another example: il y a un homme qui vous attend (there is a man [here] who is waiting for you) is noticeably different than il est un homme qui vous attend (some man is waiting for you somewhere.)

However, in classical literature and poetry, il y a was very often avoided and replaced by il est simply to avoid the hiatus, here ‘y’ followed by ‘a’ in two different syllables.

Référence: Grammaire des grammaires, Girault-Divivier, 1840, p1167.

Avoiding that hiatus is also probably a reason why this phrase is still often used in newspapers, for example an article in today’s L’Obs starts with:

Il est des meetings qui comptent. Pour François Fillon, celui de dimanche est de ceux-là.

In the context of your question “Il y a” and “il est” present the same meaning in the stylistic sens.

However, the first belongs to the current language while the second appears to belong to more refined language

In a sense, you could make a case that il est is more logical than il y a: il est en lui une vertu dormitive, for exemple, refers to the existence of a vertu, and existence would naturally involve the verb être rather than the verb avoir. But usage ended up favoring il y a over il est, and il est is not used when speaking anymore. Even in writing, it sounds quite formal.