I think most everyone has experienced what you describe - but I think your interpretation of what she's doing is skewed. She's 3. Toddlers barely play with another child (they are much more into parallel play than real and sustained interaction). They bounce from one toy to the next, one activity to the next - unless, of course, the parent wants them to stop what they're doing, in which case they have a meltdown of some sort. You are ascribing to a 3 year old such emotions and actions as "spurning," "brushing aside," and "disdain" yet she's far from capable of doing that! She's also, at 3, not capable of a mature emotional bond of the level you seem to yearn for. You talk about what you've "always dreamed of" as if it's her job to measure up to your expectations - and if she doesn't, because she's only 3, you interpret that as your failure as a mother. While you say you are calm and balanced (which I'm sure you are when things are going well), you also say a simple toddler action makes you feel "so awful" and "blindsided." That's not a balanced or calm reaction, and while you feel you are covering it outwardly, you probably are not.
You feel you need to tell us that you want her and your husband to have "a relationship" but still, there's a piece of you that's competing, or at least seeing their relationship as a threat. What children need to learn is that love expands - there's not a finite amount that has to be divided among the available adults, with each additional person somehow diminishing the other people's "share." Change that narrative! Your daughter will be a stronger and more secure person if she sees that love comes from, and is given to, many people. People who have 2 or 3 or 7 children don't love the first one less, after all! So your job is to encourage, with every fiber of your being, healthy and loving relationships with her father, her grandparents, your best friend, any nanny/sitter she sees often, and anyone else in her circle. She should not be required to hug and kiss people - be careful there! Her body is her own - but she should feel loved and protected by trusted adults. This is better for everyone, but long term, you don't want your daughter to think that only one person can love her - that's what sends young women (teens and adults) into the arms of abusive and dominant men who say, "I'm the only one who loves you and no one loves you more than I do."
Don't ask her to feel empathy for you being "alone" - she's 3 and does not yet have anywhere near the emotional maturity required to feel empathy for anyone! It's why teachers don't require children to say "I'm sorry" - because they aren't sorry! They can (and should) be required to stop doing what's wrong or to give back the toy they took, but not to express an emotion they don't feel, especially when they then learn that "I'm sorry" means "It didn't happen and it's all fine" or "Do-over" when it doesn't mean that at all.
Now, modeling this love is what you and your husband need to do. You need to be happy (and say so) when she loves her daddy, and he needs to be happy when she loves you. Take a picture with them together, and show her how much you love it! Let him do the same for you. We also did family hugs and group hugs a lot! When she watches daddy hug and snuggle with you, she learns how to do it herself. Including her teaches her (without you saying a word) that there's room for love for all. That goes a long way toward letting her know that she can have lots of friends and doesn't need to (and shouldn't) focus on having a best friend. And it's okay for you and daddy to hug without her involved too - don't force it. You want her to respect your relationship with him as well, and have enough confidence to, say, go to sleep alone in her room even though you and daddy are together. See where I'm going with this?
Please try to get to the bottom of your insecurity and fears, and deal with them. Your relationship with her will always be wonderful if you give her the freedom to grow up and learn from you, rather than be required to take care of your emotions. The surest way to drive a wedge between yourself and your child is to require her to do more to take care of you.
Please don't lament the "loss" of a close relationship when you don't really have one yet - not because you are doing anything wrong, but because she's 3. If you put too much emphasis on these small slights, you will never make it through the challenges of parenting and certainly not the hormonal phases of adolescence when they really do things to push your buttons and let you know they've ticked you off. You must always be the one who is bigger than all of this, who has an underlying trust in the love you have and will always have. That is what she will learn from, over the next 15 years while she lives in your home.