As I was reading your initial post, I noticed something that got me thinking. In your OP, you couched your question in terms of 'spreading the Dhamma' rather than spreading Buddhism'. I think that can actually present a different perspective on the issue.
The Dhamma as I'm sure you are aware is not strictly 'Buddhism'. In this sense, 'Buddhism' is really a label for particular sets of ways of thinking and acting that, over the course of history, has proven to be very effective for engaging with the Dhamma.
I am not qualified to give a nuanced explanation of what the word Dhamma entails. But from what little I know, some of the qualities of the Dhamma include:
sanditthiko -- to be experienced in the here and now
akaliko -- timeless or unconditioned
ehipassiko -- which can be examined or investigated
Dhamma can also be used in a very general sense to refer to the 'way things really are'. This means that we can 'spread the Dhamma' without necessarily spreading 'Buddhism'. According to the specific circumstances that we find ourselves in, I think we can encourage people to better understand the reality of their present experience. In this sense, people who are working in, for example, healthcare, social work, counselling, education--anyone really--can 'spread the Dhamma' without necessarily evangelizing 'Buddhism'.