I wanted to ask, is there an effective and great way to archive (public or private) Discord Channel/DMs (while also having the archive itself be verifiable/genuine)?
Yes, and it's already in this very thread:
https://github.com/Tyrrrz/DiscordChatExporter
I've personally used previous versions of this, not the current one. It can export from servers your account is in and its PMs, also allows to retrieve those within a selected range of time.
If the current version works like the one I've used, once you export the messages, it will create an HTML file; in order to save all media within, open the HTML file in your browser and then "save page as", which will create another HTML file with a folder containing all media (avatars, emojis, images, etc). That file, along with the media folder is what you can then zip and send people for them to see.
What you're asking about it being verifiable/genuine is a bigger task, because all you're doing in the end is creating HTML code that can be altered, nothing prevents you from doing so. And that's true whether it's images or video, unless the source is verified directly (like you inviting others to the server to see it) then all can be manipulated.
The reason why Wayback Machine and archive.ph (for example) work fine is because they're somewhat reputable sources, nobody
in principle is going to doctor it, or at least it's assumed, so unless a Wayback Machine type of website starts scraping servers like this and be reputable enough to be trusted, then the integrity of the data is at risk.