Aside from the moral problem of stealing bandwidth from an organization I love, would serving a unique file from the Wayback Machine serve the same purpose, and provide the same benefits, as serving that file from a CDN?
No. In fact the Internet Archive is not a CDN. Following your own link we see
A content delivery network, or content distribution network (CDN), is a geographically distributed network of proxy servers and their data centers. The goal is to provide high availability and performance by distributing the service spatially relative to end users
But while the Internet Archive has backups of its holdings, it does not have a network of proxy servers or distribute its service spatially to end users. Their site is also not optimized in the other ways that a CDN is optimized.
If the goal is to use it to offset the bandwidth usage of a site, it would be highly unethical to misuse their service in that way, and also failure-prone, since if they discovered improper use of their servers, they could easily ban your site and disrupt your service.