What is the public benefit, the public good, of rural high speed broadband?

Something about this puzzles us:

Rural areas with slow internet speeds are in line for ultrafast broadband due to a £22.2m government funding boost.

The government's Gigabit Broadband Voucher Scheme subsidises the cost of building gigabit-capable broadband networks in remote areas.

We understand the benefit to the users of said high speed broadband. But that is, by definition, a private benefit. For the rest of us out here, us taxpayers, to be paying for it there has to be a benefit to us. Some public benefit, some public good perhaps, derived from people in the boonies being able to download cat videos.

It is rather difficult to see what that is. It is as with electricity supply, or water, sewage. It is possible to be so far out in those boonies that you, the individual living that rural lifestyle, must pay the extra costs of connection to the network. We all agree that electricity, water, they’re great things for an isolated cottage to have. But the benefit flows to those who have the electricity, the water, therefore they are the people who should pay for it. So too with broadband.

One response is that the costs are so high that people won’t pay to be connected. At which point if the people who benefit don’t think the thing itself is worth the cost of its provision then why should other people pay? We’ve just gained that proof perfect that it’s not worth doing. If it costs - say - £10,000 to connect you to the internet, you do not think being connected to the internet is worth £10,000 then we’ve just proven that we shouldn’t pay - whoever pays - £10,000 to connect you to the internet.

Other than buying a few - and it is clearly few because the very point is that the boonies are lightly populated - votes what actually is the justification here, the benefit that means we all get landed with the bill?