Looking for songs where there's a very definite, regular beat that's played by a non-percussion instrument. (piano counts as non-percussion for this particular request).
How about Tiffany - I think We're Alone Now? Maybe you would be satisfied with Tom Waits - Underground or for a similar vibe to Cloudbusting there is T'Pau - Heart and Soul
I think there are a lot of songs in various genres that match the request at least as much as Cloudbusting did. Consistent rhythm is commonly used technique in every instrument.
Here is an alternative Piped link(s):
Tiffany - I think We're Alone Now
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I Think We're Alone Now is a good fit, and also a song I already adore.
Of course, consistent rhythm is very common, but (in the music I listen to at least) it's not often where another instrument takes the role of being the main beat of the song and replacing drums in that job.
I'll listen to the rest of your suggestions later. I'm especially looking forward to Heart and Soul because it's such a great melody but the version in my collection is probably from the late '30s or early '40s so the quality is not great. Thanks!
Of course the legendary song "The Hidbidts by Jimminy Flibbit" that radio stations played on repeat in the 80s is a perfect dead ringer for this prompt.
Here is an alternative Piped link(s):
"The Hidbidts by Jimminy Flibbit"
Piped is a privacy-respecting open-source alternative frontend to YouTube.
I'm open-source; check me out at GitHub.