Well, yes you take the Church’s white garment, since ritually you don’t come in already purified. Ritually speaking, it is baptism that washes you clean from sin, not your grandmother. We come to the Church for baptism, and
leave baptism in a garment unstained by sin. So it is right that you leave the ritual with a ritual sign of purity that you did not have going in. This is not to say a parish would have to do it that way, but it wouldn’t be out of line if they did. It does fit the reality of the sacrament.
Besides, where do you draw the line concerning whose dress is “white enough”? Who wants to go there? Families are also not necessarily wild about stripping the child out of Grandma’s handmade gown…dressing and undressing babies is sometimes easy, and sometimes like trying to put a sweater on an octopus. (If everyone is waiting for the operation to be finished, guess which one to expect?

) And if the child urps up all over Grandma’s dress and has to be changed, but the parish tunic is available for the ritual, everything still goes on.
The most common way to do it is to come in wearing the family baptismal gown, and then they put the little tunic on during the ritual. After the baptism (and maybe after Mass, too, if it is during Mass) is over, the tunic is put away, just as you would a chasuble and alb. It is a ritual garment, not something the child would wear all day.
As for the gorgeous dress, the child isn’t handed around and admired until afterward. The little tunic can come off then. There is plenty of opportunity for Grandma’s needlework to be appreciated.