Here is an example from archiving nytimes home page:
https://archive.tesoro.io/665dbeab57a4d57d8140f89cfedc69b5
If you look at network traffic (domain in devtools), you'll see that only a small % is coming from archive.tesoro.io -- the rest of the content is loaded from the live web. This can be misleading and possibly a security risk as well.
Not to discourage you, but this is a hard problem and I've been working on for years now. This area is a moving target, but we think live leaks are mostly eliminated in Webrecorder and pywb, although there are lots of areas to work on to maintain high-fidelity preservation.
If you want chat about possible solutions or want to collaborate (we're always looking for contributors!), feel free to reach out to us at support [at] webrecorder.io or find my contact on GH.