As you want (insist) to have working hardware on your TOSH laptop with PureOS that includes non-free firmware you need to read carefully this chapter (second paragraph) about Debian’s /firmware folder. That’s why I told you to install Debian (link above to amd64-DVD-1.iso, one option would be to replace MINT and compare two /firmware folders, one with all HW working Debian automated installation and the other with only partially working HW after PureOS installation).
Besides, I thought you were already asked to post here following output (print of some errors in red / missing .bin files) from PureOS:
sudo dmesg (like = firmware: failed to load) and
sudo lspci | grep -i wireless,
yet might not choose above explained approach (but still lessons to be understood/learned) to solve this issue with non-PureOS HW and do an install with optional step-by-step method (includes needed non-free packages).
As you are having now working Ethernet connection (Internet from within PureOS) let us go back to another experimental work(around) by using
aptitude package manager:
First you need to edit (as sudo/root)
/etc/apt/sources.list and add following line(s), first one isn’t optional:
deb http://ftp.ca.debian.org/debian/ stable non-free
deb http://security.debian.org/debian-security buster/updates non-free (optionally)
sudo apt update
sudo apt upgrade
Now you might download debian-archive-keyring (optional, not necessarily).
sudo dpkg -i debian-archive-keyring_2019.1_all.deb
Finally, execute following:
sudo apt update && sudo apt upgrade
sudo apt install firmware-linux-nonfree
sudo apt reinstall firmware-realtek
sudo update-initramfs -k all -u
Afterwards you might be able to:
sudo apt install firmware-whatever-needed in order to avoid
W: Possible missing firmware error when running:
sudo update-initramfs -u
Also (still helps somewhat), very similar not connect Wi-Fi issue that wasn’t resolved: https://tracker.pureos.net/T701, but you might persist on trying to resolve yours.
And, I admit that your late in the night questions were for myself pretty much out of the blue.
P.S. By the way (sorry),
hostapd is absent in PureOS repository and therefore leave it aside, please. And, if not used at all (from your side):
sudo apt purge cryptsetup-initramfs (but it might be linked to some other packages, who knows ).