JOEL - For one thing the church spends quite a bit of money helping out the poor in other ways; providing food and supplies for people who are suffering near home and across the world.
Also the church already spends a great portion of its money taking care of its own church members through the Church Welfare program, thus greatly reducing the burden to the established homeless programs, that might otherwise be overwhelmed by poor and homeless LDS church members.
While it is true the LDS Church has no formal program for the homeless in Salt LakeCity, the The Church's LDS Foundation has been a significant contributor of cash, supplies, and volunteer service to already established programs, such as the Salvation Army, Saint Vincent De Paul Center soup kitchen for the homeless, the Salt Lake City Homeless Shelter, the Winter Overflow Emergency Shelter, the YMCA, the YWCA and Catholic Community Service.
One example is the Good Samaritan program run by the Catholic church, where the homeless and hungry can go to the Cathedral of the Madeleine each day for a box lunch. The LDS Church provides most of the food, while Catholic church members put the lunches together and distribute them.
Another is where the LDS Church supports the Catholic Community Services by contributing food and organizing LDS volunteers to serve the lunches at St. Vincent de Paul once a week.
From what I understand the organizations already running homeless programs would like to keep things the way they are; accepting donations and volunteer service from the LDS Church, rather than having the Church come in and take over that particular area of community service.