Church schools may have their own admission criteria which include things like church attendance. However, they are bound by the Admissions Code just as much as the LA schools, so they are not allowed to take into account anything other than the factors mentioned in their admission criteria. They are also required to set criteria that are clear and objective.
Many church schools use appeal panels supplied by the LA so they are generally no more flexible on appeals than LA schools. Some (usually the ones that supply their own appeal panels) are distinctly less flexible at appeals! I don't know if there are any figures but overall I would doubt that there are any significant differences between church schools and other schools in terms of appeal outcomes. Of course, most primary schools now have their admission number set so that any appeal will be Infant Class Size, which means most appeals will fail.
I agree with Cory that advice from "insiders" should be disregarded. Even head teachers are guilty of giving bad advice on admissions. No-one at an LA school has any involvement with the admissions process at all and my own experience suggests that most teachers and admin staff at schools do not understand how admissions work. Schools that are their own admission authority (most church schools, foundation schools, academies, etc.) have some involvement in that they are required to place the applicants in order according to their admission criteria, but the bulk of the work is still carried out by the LA.
I would be amazed if MrsShrek got any priority by stating childcare as the reason for her choice of school, assuming it happened in the last few years. If she did I would encourage other parents applying for that school to appeal! As Cory says, childcare is not part of any admissions criteria. Childcare doesn't even make a good argument for appeal. The same is true of friends, transport difficulties, etc.
The Admissions Code is clear that admission criteria must be clear and objective. There is no room in the admissions process for subjective judgements as to who is more deserving based on the reasons given by parents.
If you look at the information provided by your LA, you will find the admission criteria laid out clearly. They will be entirely objective based on things like looked after children, special medical or social needs, siblings and distance. That is what is used to determine who gets admitted to each school. The only subjective judgement that has to be made is when parents argue special medical or social needs, in which case the authority must decide if the argument is justified.
So, as someone who, for my sins, knows a lot about school admissions, I stand by my original advice that unless you are giving evidence that would put your child into a higher admission category you might as well leave the reasons blank. It makes absolutely no difference to your child's chances of getting admitted.