> However [alternatives'] pages are laden with ads, have a poor UI, and you can't know for sure if they really are actually encrypting / deleting anything.
There has been a private zero-knowledge pastebin for years: zerobin, forked into privatebin now (https://zerobin.net/). In fact the original is almost 10-years old but hasn't displayed a single ad, has been open source the whole time, has had time to iron out a lot of bugs and has a better UX than this: it doesn't need cookies to work.
In terms of security, zerobin encrypts/decrypts everything in the browser. The server has no knowledge of the content. In contrast, burnernote receives the content _and_ the password in cleartext and encrypts/hashes them.
Unfortunately, as long as there's a server you will have to trust it. In both cases you have to rely on something other than code and HTTPS to know if your content actually remains private or not