organic chemistry - E1cb reaction conditions

I'm wondering why an E1cb reaction needs a poor leaving group like -OH attached to one of the carbons.

I thought the conjugate base of the leaving group must be stable enough not to de-protonate one of the newly formed alkene's H's. But if this is the case, why can't a good leaving group like Br be used in E1cb which would neither de-protonate the alkene's H's.