Given the vast scale of the Web, crawling prioritisation techniques based on link graph traversal, popularity, link analysis, and textual content are frequently applied to surface documents that are most likely to be valuable. While existing techniques are effective for keyword-based search, both retrieval methods and user search behaviours are shifting from keyword-based matching to natural language semantic matching. The remarkable success of applying semantic matching and quality signals during ranking leads us to hypothesize that crawling could be improved by prioritizing Web pages with high semantic quality. To investigate this, we propose a semantic quality-driven prioritisation technique to enhance the effectiveness of crawling and align the crawler behaviour with recent shift towards natural language search. We embed semantic understanding directly into the crawling process -- leveraging recent neural semantic quality estimators to prioritise the crawling frontier -- with the goal of surfacing content that is semantically rich and valuable for modern search needs. Our experiments on the English subset of ClueWeb22-B and the Researchy Questions query set show that, compared to existing crawling techniques, neural crawling policies significantly improve harvest rate, maxNDCG, and search effectiveness during the early stages of crawling. Meanwhile, crawlers based on our proposed neural policies maintain comparable search performance on keyword queries from the MS MARCO Web Search query set. While this work does not propose a definitive and complete solution, it presents a forward-looking perspective on Web crawling and opens the door to a new line of research on leveraging semantic analysis to effectively align crawlers with the ongoing shift toward natural language search.