@Dylan Thank you for your help! It really helped me get a better understanding of the bigger picture in regards to all this, and it helped me kinda jerryrig my own solution to it.
Thing is, shortly after installing the OS I had also decided to update truenas scale, which I found out later that it isn't recommended to do so. It didn't break anything for me at least, so I just figured it's all good.
But after having wrestled with the guides you sent me, and stumbling upon a random reddit thread, I ended up finding out that the issue causing this whole mess was that I had basically done a "default installation" of pretty much everything in Handbrake when doing it via truenas scale.
So part of what the latest version of electric eel (truenas) did, was change how the apps are handled - and how they are placed, which is more relevant for this - which messed with how handbrake sets up storage paths to "watch" and output from how it's installed.
The people in the reddit thread were talking about how it's essentially a "1-click solution" to change the already-existing apps to fit that update, but they didn't really talk much about when installing from scratch.
Which is where that guide by Theo about radarr and sonarr really helped out, because it made me familiar with the finnicky nature of the storage options and premade datasets when installing something.
So this is what solved it:
(I had already deleted handbrake quite a while ago, so at this point it's without the app installed)
I made a dataset for handbrake, in /mnt/HDDs/Applications/ with permissions set to user, group, and other, with root and read/write/execute (I figured why not, I didn't really feel like taking it one permission at a time and I'm fine with leaving it as it is now, but up to you (the reader) if you want to be more cautious about it).
I did not make a config sub-dataset like the sonarr/radarr guide suggests for sonarr/radarr.
I then installed handbrake with mostly the default options still as they are. However, I left the config storage to iXVolume because I figured it's not something I need to be messing with anywhere other than in the UI for the app and there it was already working fine previously - and then I changed all of the other storage options to Host paths with a folder I've set up myself.
(I should've made more folders for it in order to separate the pre-encoded from the re-encoded video files, but I just wanted to get it over with to see how it goes.)
So I gave it a shot, and it works fine now. Well, kind of.
I still can't see the HDDs folder, although I could at one point during my trials and errors but then it wouldn't give me access so I decided to try something else.
But now the built-in view of "storage" in the app's UI can see the video files just fine, in a place where I can place them; because Applications isn't accessible (nor viewable) to place files in, which is where I'm guessing Handbrake had assigned as its "storage" pathway before.
Anyway, lesson learned without too much harm done, that I'm gonna have to mix things up a bit to work with the updated electric eel version of truenas scale, since HexOS is still meant to be used with the version it came with during its installation.
Thank you for the help! It did push me in the right direction.