The problem with human beings is we are all different.
Half the people might want (and need) summary-only content. But many people do not trust the editorial spin that most news outlets put on their summaries.
I agree to some extent that a subsidized news media has value (though this needs to be entirely divorced from any kind of control, because even if the government tries to be hands-off with it, if there is even the possible threat of pulling funds, the news outlets start pulling their punches and failing to hold the government to account. Governments are very bad at this, and benefit greatly from doing this badly)
The way the current system works though, and I think what you are saying is, currently those summary-only people are being subsidised by those who are willing to read the full articles (and endure the adverts)
This in itself provides incentives for the publishers to provide low-quality summaries (think clickbait, but many other techniques exist) to force users into their site. For those who cannot deal with the full articles (e.g. eyesight impaired, language barriers and attention deficit), this puts them at a disadvantage, and leaves them more susceptible to misinformation and half-stories.
Aggregators, on the other hand, provide competition, which may be enough to ensure some baseline quality.
So I agree that the route you are proposing has merit; but also there is danger in it, and unless the government is going to stipulate fees that are commercially viable for the likes of Google, the net damage outweighs the gains.
Also, I've not seen anything that tells me this is what the government are doing or that it is even part of their game-plan. But then, maybe it's just not included in the summaries?