Question:  If God chooses those who will follow after him and those who will accept the Holy Spirit into their hearts, then why does He leave the evangelistic duties for Christ followers?

He does this for the same reason He gives us choice in our response.  God’s sovereignty (His control) does not rule out man’s responsibility.  Man is responsible for responding which includes man’s responsibility to share the Gospel with the lost.  God’s electing does not negate man’s responding.