How to download any iOS manually?

I'm still stuck on iOS 7 because iTunes and the blessed updaters don't work.

iTunes will start the download, but always drops the connection before finishing. It appears to be incapable of resuming an interrupted download.

I rely on an unlimited mobile data plan from a carrier that doesn't push OTA updates, so my device will only self-update over a WiFi connection that I don't really have. Routing the connection by LTE => iPad => USB tethering to Mac => internet sharing => WiFi device-to-device network => back to iPad sort-of works occasionally. Indeed it's how I got iOS 7, but it's way too flaky now.

Is there any possibility of using a sane, stable HTTP client to get the file?

I've had the same issue, when I had connectivity issues - you're right, it never does appear to resume.

You can get the update file directly from https://ipsw.me
this site merely links to the correct location at Apple for the file you require & so is safe to use

Get the latest version for your device.
This is not a way to downgrade your iOS, merely to fetch the current one by an alternative method.

If you still have difficulty resuming, then use a separate downloader app, SpeedDownload for Mac or Wget for Windows, etc.

Launch iTunes.
Plug the phone to your computer.

Then, in iTunes, hold Opt [alt] when clicking the Check for Update button & a file-picker will open instead. Find the file you just downloaded & it will run from there as normal.