I don't think lace is bad in and of itself. I was a C cup when I started secondary, with a narrow back, so I needed 'proper' bras. Actually a woman in a very 'old fashioned' department store measured me and took my mum to one side regarding 'soft cup' or t-shirt bras. (Not in front of me, but she told me later).
Most of my early bras were lacy, some were black and yes, I did get comments in the changing rooms at school. I am big of boob though and at 34, having worn well-fitting, good quality bras from a young age, I am doing pretty well in terms of sagging.
I don't have DC, so not sure what I'd do. Obviously if she's booby, a correctly fitting bra is a good thing and actually I prefer lace than anything stretchy. If the bra was plain white of plain black, might be just fine. One thing I recall is that my mum practically never bought me matching sets (couple of times when the pants were very conservative, maybe). I actually think that's a good thing as bra was (yes, cute, because I wanted to look nice in the mirror) primarily functional, even if I wanted it to be pretty.