Cloudflare not forwarding ETag header?

I spoke with a Cloudflare Tech Support Engineer and he helped fixed my issue. For anyone in the future who might have this same problem, here's what fixed it:

I had Email Obfuscation enabled for the URL in question (that was getting its ETag header removed). According to https://support.cloudflare.com/hc/en-us/articles/218505467-Does-CloudFlare-support-ETag-headers-,

Email Obfuscation will need to be disabled, otherwise the ETag headers
will be removed from the response. Email Obfuscation modifies the code
significantly enough it cannot be considered semantically equivalent.

Disabling Email Obfuscation still didn't fix it, though. Turns out I had an app enabled that would modify the HTML, so the ETag had to be invalidated. In the end, setting a Page Rule with Disable Security, Disable Apps, Disable Performance for that URL did the trick.