Well, advertising is usually billed to advertisers as a cost-per-thousand (really, cost-per-mil, or CPM) for impressions. For a small local news site, they'd be super happy to get a $10 CPM on a local online ad. So let's just say for fun that we'd get the same rate for the hypothetical 'Spotify for news' service. Say a locally focused article about the latest city council meeting gets 20,000 views. At a $10 CPM that only generates $200 (or $0.01 per person). Does that seem like enough revenue to cover the time it took the reporter to attend the meeting, write up the story, have it proofread/edited, and then posted online and printed and still make a profit? Probably not!
The sad part is that the $.01 per person rate is way, way higher than what Spotify apparently pays out (around $0.006 to $0.0084 from some Googling).
This starts to look better if you just go to something like $0.25 per read, but at that rate you're likely charging much more than it would cost to buy a print copy, or a per-day subscription rate, so i'm not sure the economics would work out there either.