I think the trouble here is the 'eventually I give up' thing, because things won't change until you're consistent. It must be confusing and frustrating for her that she knows she'll end up in bed with you eventually but you don't immediately respond to her 'request' for that to happen, so she's crying out of frustration that you just don't 'get' what she's saying, and is only able to calm down when her 'need' to be there in the big bed with you is met. You and your husband should have a discussion about the different approaches, agree on one and stick to it like bloody glue.
Possible solutions:
She shares the 'family' bed, or
You and she sleep together in the spare bed
(she might grow out of this in a year or 2, and be happy to have a 'big girl bed' of her own, or she might be 11 and still want to be in with you, it's a gamble!)
You do away with the cot and invest in a Montessori 'floor bed', in the hopes that she'll feel less trapped and more independent/confident, and therefore less anxious without you
(unlikely to help at all, but you never know! Obviously you'd need to completely baby proof her room too)
You commit to a program of controlled crying
(can be deeply upsetting for you as parents as well as her. May work like a charm within a few days, or may drag on for a week or more and leave her with general anxiety and clingyness during the day even if she gives up crying for you at night.)
You adopt a method of 'gradual withdrawal' - when she wakes you sit by her cot and 'ssshhhhh' and pat or whatever but don't take her out, and as she gets used to falling back to sleep like this you move your chair further away each night until you're out of the room and she can settle herself
(can be confusing and frustrating for them. Likely to take longer than controlled crying and be as upsetting for her, but maybe not so traumatizing for you)
In your shoes I think I'd be tempted to go for the gradual withdrawal, but have your husband be the one to sit with and soothe her. If she associates you with being the one who gets her out and brings her into bed she'll quickly learn that that isn't how it's going to go with Daddy, but still have the reassurance of having a loving parent in with her. I'd give her a top you've been wearing to snuggle down with when you first put her down for the night too, so she can be calmed by the scent of you.
The most important thing of all is consistency, no chopping and changing, even if you're all exhausted - you need to give whatever method you choose time to work, and that could be a month or more!