A church follows its leaders. If you want your church to be evangelistic, they must see and hear your burden for non-Christians from you personally. Sometimes that can be done in the simplest way.
For example, consider the Sundays following tragic events, or perhaps even Sundays near the anniversary of a tragic event such as 9/11.
These times may present an opportunity to say something like this:
“Many of us vividly remember the events of September 11, 2001, when al-Qaeda hijacked four airliners for suicide missions against our country. We are one of the wealthiest countries in the world with a large budget for defense, but we could not prevent the attack. Life is not guaranteed.
James 4:14 reminds us: “You do not know what will happen tomorrow. For what is your life? It is even a vapor that appears for a little time and then vanishes away.” That is why it is so important to share Christ with anyone and everyone and do it today. This week I'm asking God to give me opportunities to share my faith. How many of you would do the same?"
Regardless of how your service is structured, it takes less than two minutes to say the above and keep evangelism in front of the people.