There are a whole range of reasons for tentmaking but I don't think evangelism is one of them. Well it might be. But it's not one of the reasons given to us in the Scriptures. The apostle supported himself financially because...
- he had to out of necessity
- he wanted to as his 'boast'
- he was setting an example to young Christians of hard work
- he sought to distance himself from mercenary philosophers
But not because he wanted to have a more authentic connection with the unbelieving world.