When I'm worker stealing, I'm usually not trying to choke. Instead, I'd like the AI to build more workers for me. So after stealing a worker on the DOW turn, I try to distance myself from the city: if you are more than 2 squares away from the city square, they do not feel threatened and don't feel the need to switch to spamming archers. And if a unit comes chasing my warrior, I just run away. Since I already have some war success from stealing the worker, the AI will eventually make peace, much quicker on difficulties below deity. So I wouldn't consider spending 80 gold on an axe which won't really do a better job of worker stealing compared to the warrior.
Eventually (and quicker if you let the AI spam units), they will begin to escort their workers, at which point you do probably need a stronger unit like an axeman, but I'd much rather whip out a couple axes than spend 80g on an upgrade.