In my choir there are only one or two people who have "lead singer" voices. But I can usually find some song that matches each person's abilities. Some songs have spoken leads, or shoutin'-and-hollerin' type leads, and those can work for people who are not songbirds.
Honestly, if someone goes flat a lot, I would rather have them do a lead than have them in the choir mix messing up the harmonies.
If a person is fixated on one particular song that they want to lead and it would just be a disaster, I would tell them "that song isn't the right style for your voice, but I think the lead on _____________ would be a good fit for you."