Ordnance Survey map suggests that the trackbed to Wrington is intact, but not only would a further section require significant earthworks/tunnelling, but would be a long way round from Bristol, and without a tunnel under the airport itself, end up on the wrong side for the terminal buildings; so a route from the north would seem more practical. In any case, we are still talking about a link purely from Bristol to the airport. so the Devon Live speculation about long distance links from the south-west seems spectacularly wide of the mark.
I think that it rather depends on what you want to achieve (and how much you have to spend on it).
If you want to get people from central Bristol to Bristol Airport faster by public transport at minimal expense, the most obvious solution is to extend MetroBus, possibly with the addition of a few new bits of roads and bus lanes from the A38/A4174 roundabout. Even if the airport offer to pay for it, there's no way MetroWest is going to see their
BRT▸ baby eclipsed by a tram, and good luck with trying to figure out how to diverge a heavy rail line from the
GWML▸ and get it up the hill to the airport.
As has been suggested though, if you see heavy rail as enabling the airport to be better connected to the wider region, then an approach via Yatton and Wrington isn't necessarily a problem, and the fact that you could easily have a south facing connection at Yatton and possibly even have several platforms for 8-car trains under a new terminal building could be very appealing indeed.
Yes, if you look at the journey times, then 18 minutes from
BTM▸ to Yatton, plus say 5 minutes up the hill to the airport, is more or less the same as the bus. But it's about 15 minutes faster if you live in Bath (assuming a direct train) and half an hour faster if you live in Weston. And of course you are sitting in a train rather than on a bus (or in your car) and if the line were electrified and resignalled, then the journey times could be significantly improved.