How do children learn the meanings of the simple, concrete words that form a majority of their early vocabulary? Statistical information about many word meanings can be recovered from the gradual accumulation of information about the co-occurrence of words with their referents, a proposal known as ``cross-situational learning''. Contrastive vision--language models allow this learning strategy to be tested on egocentric video data capturing children's early real-life experiences with visual and linguistic inputs. Initial successes of this method on small datasets provide a proof-of-concept, but vision models struggle to learn robust visual representations when trained on these same egocentric videos. How dependent is cross-situational word learning performance on the quality of the model's visual representations? And can cross-situational models scale to larger and more diverse datasets? To address these questions, we trained models on the BabyView corpus, including 2,441 hours of longitudinal egocentric video from 48 children. Models trained on this dataset learned object--label mappings, but performance was far higher when they used internet-pretrained visual representations as opposed to when visual representations were learned solely from the developmental video data. Further, the majority of the learning signal was carried by a small proportion of utterances that described a visible referent. Models learned lexical representations that showed recognizable human-like semantic structure, but model performance showed substantially different trajectories over training experiences than those observed in children. Cross-situational learning is a viable word learning strategy for early noun vocabulary, but constructing developmentally plausible models will require understanding how to learn high-quality visual representations from naturalistic data.