Having been to Tristan da Cunha myself I can assure everyone that there is absolutely no chance of building any sort of airstrip on the remote island which is one of the remotest islands on the planet.
The islands are more or less covered in cloud for the majority of the year and the sea conditions would definitely prohibit any sort of sea plane service.
The island is served on a very irregular basis by a small cargo vessel (3 times a year) since the withdrawal of the RMS St Helena.
There are only about 250 residents on the island (mostly inbred) and the cost of such a strip would be very costly indeed especially considering that there is no available or suitable land on the island to actually build one.
For those proposing this idea and especially to the OP I think that you would actually need to visit the island to appreciate the situation.
On St Helena they had to take away half a hill to build the airport and even that has caused issues.
So in short I'm afraid this is an absolutely no go.