This is something I am currently wrestling with. What strikes me is that every Christian has far more knowledge than Ezekiel had. We know what the plans of the the Lord are for the future of every individual we meet - that he has appointed a day when he will judge the world by the man he has appointed, and that he has given proof of this to all men by raising him from the dead (Acts 17). And we know that the only hope for anyone on at that day is to come to Jesus as Lord and Saviour now before we meet him as our judge.
It is correct that there is little in the New Testament setting out a positive obligation on Christians to evangelise and tell people about the coming judgement. However, that is not as surprising as it might seem because the Bible of the early church was the Old Testament. It was the Old Testament that they went to for training, correction and rebuke so that they were fully equipped for every good work (2 Tim 3), and the Old Testament through Ezekiel sets out that those who have knowledge from God about the future destiny of a person are responsible for sharing that knowledge.
So, it seems to me that the obligation Ezekiel had to warn people to repent and believe is an obligation that every Christian has, and like every obligation that Christians have, we need to bring our daily failures to carry them out to the Lord, receive his forgiveness and ask for strength to keep going.