Not really. The most prominent case where that happened is a shiny new port they built in Sri Lanka, but which turned out to be impossible to run profitably, so the Sri Lankan government handed them the port for a 99-year lease instead of paying their debt. Now China has a port in Sri Lanka they can't run profitably either.
What the BRI is is a great opportunity for Chinese officials to earn kickbacks from Chinese companies for helping them land government contracts abroad. I assume that's what happened here: someone working for Huawei really wanted to close this deal and receive a fat bonus, so they offered the ambassador a cut for intervening on his behalf.
So whether the primary aim was economic or military doesn't matter much, since the outcome was an expensive failure either way.
China builds you a new airport, and suddenly you are indebted to them for decades.
As for the Faroese Islands, and in particular Greenland, they don't care so much about their economies. What they care about is their proximity to the Arctic ocean, and its potential riches. Having a foot in the door, would be a huge boon for China. Indeed, the notion of China building oil rigs and mines in Greenland has been a subject debated a lot in Denmark.
Pressuring the Faroese may be a good way to make Greenland follow suit. But there I am merely speculating. Besides, the Faroese are the ones making overtures to China, so China can counter with demands like this at no expense to themselves.