The device does not get a new version of iOS, so I tried Safari to see if the web client works. Spotify's web client does not work on that version of Safari.
So I tried downloading Firefox and Chrome, but I can't because they no longer support my version of iOS.
So I looked for a way to download older versions of apps (Spotify, or an older version of Firefox or Chrome on which Spotify would still work). As it turns out, there's a way: you first need to "purchase" the app you're interested in on a newer device that supports the latest version. Then, you can log into your older device and download an older version of the same app.
That means there's a way for me to run an older version of an app that works on my iPad, but nah, I need to first get a newer device and "purchase" it there.
My conclusion: the onus is on the device manufacturers to not arbitrarily restrict their users. Jim Nielsen's mom's iPad is perfectly capable of opening that website, but its manufacturers are limiting it artificially.