No. The zone file lives on the nameserver (DNS server) which Route 53 provides. Namescheap registers a list of nameservers for your domain with the TLD.
An over simplified example for looking up "news.ycombinator.com":
1. First query the TLD nameserver for all ".com" domains asking for the authoritative nameserver(s) for "ycombinator.com".
2. Next query that nameserver for "news.ycombinator.com".