I’ve heard a lot of people in the past use Luke 10:1-2 to support the idea that we should all do "cold contact" evangelism. When I look at this verse, I see that it says not to "greet anyone on the road" (verse 4). Does Luke 10 really apply to cold contact evangelism?
I get the sense that you feel it is an abuse of scripture to apply Luke 10:1-2 to tell people they need to go out and "share their faith" with people they do not even know. I think that yours may be a bit of an overreaction, but my answer would depend on the way in which the passage is being used. For example if a preacher were to tell his hearers that Luke 10:1-2 is a command for all Christians to devote a certain percentage of their time to sharing Jesus with strangers, that would be a big stretch. If he were to continue to imply that if we do not we are disobedient to the command found in Luke 10:1-2, that would clearly be an abuse of this passage. Jesus is obviously not commanding all his followers to go on this particular mission. If the command here is to anyone, it is to the seventy followers.
To be honest with you, I really doubt that you heard the passage abused in this way. On the other hand, if a preacher were to use this passage as an example and as an encouragement that this is a great way to get the message out, that seems perfectly fine to me. In other words, as long as the passage is not being presented as a command, but rather as an encourageing example we can follow, I do not see it as abuse of this passage. Perhaps what you heard lies somewhere between the two scenarios I already described.
I think the real question here is whether you, as a member of a local church, ought to feel obligated to go "cold contact sharing" if one of the leaders of the local church asks the members to consider this activity. My advice is that you ought to at least consider trying to support the efforts of the leadership of your local church to get the gospel out. You should not feel obligated, as we have freedom in Christ. In my opinion, you are not in sin if you feel this is not a good use of your time and talent and choose not to participate. If you feel your freedom is being violated–that you are being told that you are a bad Christian if you do not show up, then I believe you may have a legitimate gripe. However, my suggestion is to do your best to give this person the benefit of the doubt and to try to make his work a joy, not a burden (paraphrasing Hebrews 13:7). Even if you choose not to participate, make sure that your leader feels 100% support from you for his efforts to evangelize and offer yourself to be part of the program, even if not in this particular way. If your only response is to criticize, then you are in danger of being a hindrance to the work. You will do well to support your leader, not undermine him.
By the way, I disagree with your point that v. 4 (not greeting anyone on the road) is evidence that this passage is not even about sharing our faith "cold contact." This was simply practical advice to make their mission work more efficiently. I think he wanted them to get them to the city where they were to do their work. It was not a commandment to not share our faith as we go. Perhaps you could be accused of trying to find a reverse command here when you are trying to counteract another person finding a command here in the first place. Just a thought.