I personally get very funny about other people that I don't know extremely well taking my kids in their car without me. Granted, it's helicopter parent-ish, but they are my kids, I love them.
Don't be deterred on being a FDC provider new mum. I'm not against it as such, but I'm very protective of my kids and just don't like the thought that there is only one carer. Especially with small children who can't talk, I believe there could be a propensity for covering up abuse. I'm sure only a tiny percentage of FDC workers are like that.