Something like this is certainly more convenient, but ultimately the best method is education.
And since that belief is evolutionarily beneficial, you can see how it would eventually turn into a bigger and bigger problem...
If you're thinking of Christians, it isn't necessarily all-or-nothing. Although I believe the Catholic church disapproves of all contraception, Protestants typically approve of anything that prevents fertilization. Once the egg is fertilized, it's considered a human life worthy of protection; prior to that, the egg and sperm are just cells. So this would be considered fine.
As for Protestants, you'll find many who agree with the Catholics, many who agree with what you've called typical, and many who consider human life to begin at birth. But most people have a loose affiliation with their denomination, and I don't think there's much that can be called typical about the rank-and-file Protestants or Catholics for that matter.
If people want to force you to be sterile, wouldn't they use a different method?
e.g. Offer people generous welfare if they elect to undergo this temporary birth control procedure.
[1] There are only so many ways to get to the vans deferens without surgery, so I'm not too worried about missing something obvious.
Or nanobots, I suppose, whenever we get those. That doesn't seem likely to be any time soon, sadly.