Not feasible.
First, the Falklands do you no good, they're actually due North, and East of Tierra Del Fuego. Distances range from 350 to 500 kilometers. They also don't get you anywhere. It's 1210 from the Falklands to the South Orkney Islands, and 1370 from the Falklands to South Georgia/Sandwhich. If you want to take the long route, its 2200 from the Falklands to the South Orkney's, with a stopover in South Sandwhich. Add in another 500 from South Orkney to South Shetlands, 1710 to 2710, and another hundred from there to the coast, 1810 to 2810. So really, the Falkland aren't stepping stones to anywhere. The gaps are too immense.
You might as well jump straight from Tierra Del Fuego to the South Shetlands directly, that's 850 kilometers, from there to another 100 to the peninsula, 950 kilometers.
Of course, that's as the crow flies. Conservatively, you would want to double or triple those distances, because almost certainly, they wouldn't be travelling directly. You couldn't travel directly south the 950 kilometers from Tierra to Antarctica because the winds wouldn't let you. No southerly winds at all - just the 'West Wind Drift'
You could use the West Wind Drift to travel from South America to the Falklands, and people inadvertently did. And you could use that same West Wind Drift to travel to South Sandwhich. But then you're screwed. The Winds are going in the wrong direction now, you've got to move opposite the winds to get from South Sandwhich to South Orkney. So you can't actually get there that way.
So your better attempt would be to Island hop from Fuego or Falklands directly to South Orkney. That gets you closer. But the winds are still going in the wrong direction now, to jump to South Shetland. So you can't get there that way, either.
Keep in mind that these winds are not gentle breezes. No, they're known as the Furious Fifties ("there is no law") and the Screaming Sixties ("there is no god).
I did some calculations of distances:
*Tierra Del Fuego to Falklands - 350 to 500 kilometers
*1370 from the Falklands to the South Georgia and South Sandwhich Islands
*850 from South Sandwich to South Orkney
*1210 from Falklands to South Orkney
*1660 from Tierra Del Fuego to South Sandwhich
*1260 from Tierra Del Fuego to South Orkney
* 850 from Tierra Del Fuego to the South Shetlands
* 1050 from Falklands to the South Shetlands
* 500 kilometers, South Orkney to South Shetlands
100 Kilometers from South Shetlands to the Antarctic Peninsula
Here's a map: of the Islands
Here's a Map of the Currents:
It would almost be easier for Tierra Del Fuego to colonize Africa.
I can't see how the local boats or quality of navigation could stand up to that.
When I ran the numbers and currents for Land of Ice and Mice, there were a lot more Islands, they were a lot bigger, the extreme maximum distances were 350 to 500 km, and there were a lot of local crossings of up to 5 to 100 km to cut teeth on, and an intense cultural tradition of accessing sea protein on a regular basis. A lot of the big crossings had intermediary island specs to land on.
And remember, the winds don't change direction. So anywhere they take you... it's a one way trip. You can't get home.
This is simply beyond infeasible.