Why would it need to be free? It could be an ordinary investment that's paid off over time. (Though I assume that the payback period is probably quicker than 5 years, that seems a bit too long to be useful.)
Be careful about cows: there's quite a few charities who want to give out cows to help people (using the same logic as you do). But for many recipients the cow is a net drain on their resources. That same logic applies when buying a cow with your own money.
If the Bitcoin rig isn't free then the villagers don't get shit for years while it gets paid off. Then they're just sharecropping Bitcoin.
They could invite outside miners to set up shop in the village.
The miners would finance provide the capital, set up the mining rigs, and take all the bitcoin risk. The villagers would just sell them electricity. Just like they would sell electricity to any other business.
No clue whether that's what's happening in Bhutan, but it's a perfectly valid business plan, and pays the villagers right away.
It doesn't make any fiscal sense to find some out of the way location with a small scale hydro generator to set up a Bitcoin miner. The amount the village would need to charge to make a worthwhile amount of money will certainly be above large scale utility rates.
The Bitcoin miner in remote African village story smells like someone taking advantage of some villagers. They're either paying them below market rate for electricity or indenting them with a Bitcoin mining setup that will barely pay for itself over years.
If virtually no one else would buy the stranded electricity, the market rate would be zero otherwise. Anything more than zero is better for them.
There's more than one bitcoin miner in the world, so the villagers could get some competitive bids.
Who is giving free money to remote African villages to buy Bitcoin mining rigs?
For reference a milk cow costs about $1500 and can produce 2,500 gallons of milk in a year. For the cost of the Bitcoin rig maybe getting $10 a day a village could buy about 13 milk cows. Even selling milk at $1 a gallon that's about $88 a day. The "free" power can be used to Pasteurize the milk (UHT) for extra longevity.
That same $20k could buy a room full of laptops plus networking equipment and Starlink. They could take online courses to learn a variety of technical jobs and then do contract work for way fucking more than the $10 a day they'd get from a Bitcoin rig.
Edit: fixed Antminer from autocowrong