Agree with subzero - I think part of the problem is that it is very difficult for children to settle into daycare if they're only there one day a week. It's such a huge gap for them from one week to the next. The daycares in my area will only accept children for a minimum of two days. I'm not sure if your financial situation allows it but if so, I'd seriously consider increasing her time there. Is two half days an option? - that might work better than one full day.
With regards to the teacher saying that your daughter might do better on a less busy day, I think you've taken the comment the wrong way, I'm sure she was just making a helpful suggestion rather than trying to say that they didn't want your daughter there. This is going off on a bit of a tangent but, my DD has recently been complaining of some of the boys hitting her at daycare, and the response from the Director of the centre was that yes, those boys are known to her as troublemakers and she is trying to deal with it, but my DD attracts the trouble to an extent as she doesn't cope well in a very large group so they see her as an easy target. The point being, some children manage well in large groups and some children don't, and TBH if your daughter is an only child and hasn't been to daycare before then it's likely that she is going to manage less well than some other children, so why not move her to a quieter day if you can, if it will help her cope? Even on a quiet day there will still be enough children there for her to get a good deal of social interaction.
Anyway, I don't think you should beat yourself up for being overly sensitive! It is a great quality to have! Most people would find it upsetting for a teacher to say something negative about their child, it's normal to take it as an insult towards your parenting, and of course it's worse if you have any guilty feelings towards sending your child to daycare.