[Miscellaneous Android Development] xdaillum:
Ok I managed to figure it out. Here is basically a log of how I ultimately solved this. I've also got some bugs to report so I will report that in a separate reply.

The latest version as referred to in OP (2020-04-13) is available in the LOS 17.1 folder and has a slightly different date format in the filename, but the filename is: LOS17.1_13c0420_FIH_SS2.SAT.HH1.HH6.HD1.DRG.7z - seems like no one else was as lost as me on this though LOL. I also got a bit lost because the earlier versions had "DAILY" which I took to mean "stable", then there were some "beta"s and "test" ones which I took to mean not as stable/possibly more buggy.

I tried using ADB sideload via the stock recovery, but it wasn't recognising any of the files I was trying to push through via ADB sideload. I tried pushing through boot.img, an unextracted LineageOS zip file, TWRP recovery - but all kept failing with errors along the lines of "package not recognised"

Because this was constantly failing, I decided to try and go back to the Download Mode method

I still don't know why my computer wasn't picking up my device in Download Mode with nothing appearing in Device Manager, but I tried it again with the same computer and this time it appeared

I had to re-install drivers (even I previously used this same computer to do everything on my Aquos S2), but to do that, I had to "Disable driver signing enforcement" because Windows 10 wasn't allowing the driver to be installed as it wasn't properly digitally signed or something

I obtained the drivers from here, and then had to edit in the text

;Sharp S2

%SingleAdbInterface% = USB_Install, USB\VID_0489&PID_C009

%CompositeAdbInterface% = USB_Install, USB\VID_0489&PID_C009&MI_01

under the section starting with "[Google.NTamd64]" in the file "android_winusb.inf" (edit with Notepad).

After that, with the phone plugged in and still in Download Mode, with driver signing enforcement disabled, and this time now being detected in Device Manager as an unknown Android device, I was able to successfully update the driver by pointing it at the "android_winusb.inf" file above.

I then ran

fastboot flash boot boot.img
fastboot flash system system.img
fastboot flash vendor vendor.img

from the latest LineageOS. system.img took longest - which is not surprising, considering the file size.

After that, I tried to reboot into recovery as per instructions on OP to "reboot to recovery, format data, then install Gapps". So I ran command

fastboot reboot recovery

but was getting error

fastboot: usage: unknown reboot target recovery

I also tried

fastboot boot recovery.img

but it kept rebooting phone straight into Download Mode, even though CMD showed the command was executed successfully.

"recovery.img" was the filename for a TWRP.img file for Sharp Aquos S2 that I downloaded. This is supposed to boot into the recovery mode of the recovery.img file without flashing/installing TWRP over the existing stock recovery.

I turned phone off and tried to access recovery via button combinations. Failed - so it started up as normal - but it was now on, and successfully in LineageOS! Albeit with no Google Apps (because I hadn't flashed them yet)

I turned on USB Debugging through Developer Options, then used command

adb reboot recovery

and it rebooted into TWRP (as it's included in the ROM)

Formatted data, then flashed Gapps, then flashed Magisk

Then it was basically back up and running and as normal!