Perhaps it is less effective today. Our culture has and continues to turn away from the church. And even for those who attend, many, as the pre-eminent living American theologian Dallas Willard is wont to say, are not Christ followers. I suppose we could call them “cultural” Christians.

The average American church has one person come to Christ in any given year. Depending on getting people to “go to church” to repent isn’t working very well. What works better? – Taking people into our lives so that they will then take us into theirs. This is how, as you point out, “we show people that we genuinely care about their eternal future”. Until we take people into our lives, and they then take us into theirs, they are not ready or interested in hearing about heaven and hell.

I just read a book by John Hayes that chronicles his experiences working among the very poor (very few of whom were Christians), in the U.S.A. as well as in other countries. His story of working and living among the poor in L.A. included his lament that in the several years he was there, he only led a couple of hundred people to Jesus. Would that I could say as much!