I haven't done a systematic look into whether cover songs fall within the same phase of the 15-year cultural excitement cycle as the original version, as I just did with nostalgia songs that pine for a particular cultural moment. (I have discussed examples every now and then, just not systematically.)
My sense is that they do, although the effect might not be as strong as it is for nostalgia songs. With nostalgia for a narrowly contained historical-cultural moment, you can't help but be in the same mood. But with cover songs, they're more open to interpretation, allowing an artist to take a bouncy, upbeat, carefree song from a manic phase and give it a more mellow, vulnerable, emo rendition during a refractory phase. I think it's still more natural to cover a song from the same phase that you're currently in, but there is more room for artistic license.
At any rate, until I have time for a more systematic investigation, I'll put up mini posts like this one to showcase examples.
Both of these songs were produced by the restless warm-up phase of the cycle, when people are stirring awake from their emo slumber of the refractory phase. They want to get their bodies moving -- giving rise to simplified, easy-to-dance-to music -- as well as exercise their social muscles, which have atrophied during their withdrawn phase -- marking a turn in tone toward the flirtatious.
Stylistically, this phase tends to be more stripped-down and back-to-basics, since it's the start of a new cycle. A cycle could hypothetically end after any stretch of three consecutive phases, but the prolonged emotional crash and drain, as happens during the vulnerable phase, is the most salient marking of the end of a series of cultural moments. When the energy level re-sets to the baseline, it's possible to start something new again.
The original is from the extraverted and cheerful form of doo-wop from the early '60s warm-up phase, which contrasted against the moody form of doo-wop from the emo late '50s. The cover is from the disco-punk late '70s warm-up phase, which felt nostalgia for pop music of the early '60s -- before the moody prog rock of the early '70s emo phase.
The original was a hit in the US, while the cover was only big in Britain and Europe, where the punk genre actually found chart success.
"Denise" by Randy & the Rainbows (1963):
"Denis" by Blondie (1978):