Say I'm reading a paper taken from my Zotero library which provides a really useful overview of a particular topic and I want to explore the papers it mentions throughout - is there an easy way to essentially copy and paste that paper's bibliography into my Zotero library? These won't necessarily be papers which I've already added to my library as they are papers which have merely been referenced by a paper I'm reading.

I hope this makes sense

The Reference plugin can the extract reference list from a PDF in your Zotero library. And can take you to the DOI/URL in your browser for any one that you do not already have. There of course you can choose to add/download it with the Zotero web connector.
https://github.com/MuiseDestiny/zotero-reference

Unfortunately the current documentation on github is out of date (only for the older-style, v6-compatible version). And the plugin's Settings dialog in Zotero is only in Chinese. But Chrome's inbuilt translation does a reasonable job on the github pages. And the rest is fairly intuitive (if necessary, Google Lens does a decent job of translating the Settings dialog). Someone has just added an Issue for English settings to be added (the older, v6-compatible version had that).
https://github.com/MuiseDestiny/zotero-reference/issues/355

But there's nothing in the Settings you really need to set in order to make it work (except for Chinese paper access via CNKI for those who need that). And it's so powerful that it is worth the effort to figure out how it works.

It can also show you a list of the papers that cite the current paper (with data from Semantic Scholar; for which a [free] API key is not required, but can be used if you get one). So the plugin's two display modes are References (that the paper cites) and Citations (that cite the paper).

Are you running Zotero v6 ? Because that looks like the Settings dialog for the old, v6-compatible version of the Reference plugin. The Settings dialog under Zotero v7 for the latest version of the plugin (1.4.4) looks like this, even on English language PCs:
https://s3.amazonaws.com/zotero.org/images/forums/u5906489/9icja67g6ss1v7l762ir.png

The old plugin version functions quite differently. It can still extract reference lists (but not get the cited-by list); but it does that in different ways. The new version is cleaner and more powerful (albeit with the only-Chinese Settings). On the plus side, the current github documentation (translated by Chrome for example) describes the old version. So that version may still be of use to you.

If you're OK modifying your workflow a bit, you can try a tool called RefTraceback available here:
https://sourceforge.net/projects/academic-tech-toolbox/

It's clunky but can save some time if you're researchin' hard.

Basically, it allows you to bulk-search the refs (you can extract the bibliography and select the refs that look interesting), and then you can save just the ones you need using the Zotero extension in you browser.

I've been meaning to develop this into a Zotero plugin, but have zero knowledge about how to do that. Any feedback/help welcome!