I'm in the same demo, and I wasn't aware of ellipses being seen as an habit of old people.

Biden's (or likely, his team's) usage is atypical; ellipses are usually only used when you're citing somebody, to indicate unintelligible segments or sentences that are not crucial to the thought being relayed, and when you're voicing some degree of surprise, doubt or reticence about the forthcoming sentence (e.g. "She thought I was ready... maybe I wasn't.")

(If they ever come for my semicolon, I'll take up arms.)

The example is correct usage... not perceived as old at all... but when grandpa replaces all punctuation... peppered in between disjointed fragments of thought... LOL! Very difficult to read!