This needs to be unpicked.
Yes, it is the law that a child has a right to a relationship with both parents. There is thus a presumption that direct contact with both parents is in the child's best interest.
This presumption can be overturned. But it needs to be overturned by evidence.
It will not be sufficient to overturn this presumption to say that one parent dislikes another or finds him/her annoying or unpleasant. There will have to be serious, corroborated evidence that direct contact with the parent will be harmful.
So, as others have said, stop worrying about what the 'ideal' should be - of course in an ideal world your daughter would have a healthy, loving relationship with her dad.
But lets deal with the real world. Decide what you are really worried is going to happen here. If you are genuinely worried that he would hurt her if he ever got the chance, that is a very serious accusation and you are going to need serious proof. Usually, only corroboration from other agencies like hospitals or the police or independent witnesses will be enough to establish this. If you succeed in proving this, it is an argument for no contact.
If you are worried he would be careless/inconsiderate, again gather your evidence. This is an argument for supervised contact until he can demonstrate he has adequate parenting skills.
If he is abusive and unpleasant to you, not physically violent but emotionally abusive, this can be very tricky as of course your daughter will in time pick up on it but it is going to be hard to convince a court that this is a reason for no contact or long term supervised contact.
if your current solicitor is not explaining what is going on in a way that makes sense, get another one.
I would really like to know what possible justification there is for you all having contact together. Madness.