I've tried several distros to fit on my repurposed Chromebooks that came with 16 GB emmc storage. Debian was the smallest one, using by default about 5.5 GB of data, plus 1 GB for swap, plus the boot partition. I had about 9 GB left after installing, with XFCE. After I installed a few apps and games, I ended up with 6 GB free space. It works fine and it updates fine.
i'm also using a chromebook with 16 gb emmc storage. my installs keep being about 10 gigs with xfce. i was thinking i would end up with just about your numbers but there is almost 5 gigs of other stuff i ended up with from the live disk installer.
Install the minimal/netinstall image, and then add what you need.
You'll probably spend less time adding than trying to figure out what's installed that you do or don't need and trying to remove random packages without breaking anything.
i have broken the install a few times just deleting stuff. The live disk won't find my wifi card so i can't net install unless I buy an ethernet adapter.
In case it helps: I have solved that problem for myself using a $9.00 USB Wifi dongle.
For whatever reason (other contributors facing the same issue?), I have found that every cheapo USB Wifi dongle I have tried has worked perfectly with the minimal Linux images.
I realize I might have just gotten really lucky a bunch of times, but it could be worth a try.
If you have another USB drive, I think you should be able to load the wifi drivers from that when using the netinstall. I am pretty sure I remember doing that 15 years ago.
You can start with dpigs. Then start marking packages automatically installed with apt-mark. aptitude may be a good frontend when removing a lot of packages, you can mark entire categories, like libraries, as automatically installed.
Pay attention to the package headers when removing packages. You don't want to remove essential packages.
a quick place to start would be the systemd services that get automatically started when you boot your system. when i did this in the past, i would google each service that was running to determine if i needed it and remove the associated software if i decided that i didn't.
(since you're using debian): if it's a fresh install, it would make more sense to start with a minimal install first like the netinstall image and then pick and chose what you want to put on top of it.
if your issue is that the distro is too bloated: there are other minimalist distro's out there (some are based on debian) and they've already gone through the hassle of figuring out what the bare bones minimum is for fully functional distribution that can serve a viable daily driver.
if xfce is what you want, try a custom install (using dvd1) and just pick xfce instead of the gnome default during tasksel. you will get a few desktop applications like libreoffice and firefox esr, but those are easily removed if you don't want or want to replace them. using dvd1 as my install source, wired and wireless drivers were set up during install, were available during install, and were ready to go on first boot to xfce (on an am3 pavilion desktop test system).
@Magister@WhiteOakBayou There are a lot of things to like about MX, nice interface, I really like that you can boot up using either systemd or sys-V, since systemd tends to be a lot faster but also tends to break it makes it really nice to have a sys-v fallback when things do break. Support has been excellent, I've yet to have it take them more than three days to fix anything broken I've reported, contrast that with Ubuntu where if it happens within the next three major releases you're doing good.