What I ended up finding out is that it's possible to turn a website into something that feels more like a native web application - at least it gets its own launch icon in iOS. In reality this is just a web view with a basic application launcher, but it's still a neat thing to have and is next to no effort to implement.
This relies on the Web App Manifest W3C Specification which both iOS and Android appear to support by the way. There's also a handy Web App Manifest Generator available to make things easier for you and Apple has documentation on additional customisations that are possible.
So here's how it's done, first your website (in its <head> section) must contain a reference to the web manifest file, something like this...
The manifest file is a JSON file which can either be hand crafted or created via a generator such as Web App Manifest Generator. In my case this is what I ended up using...
The scope member is defined to address an issue with iOS Safari opening links in a new window when it shouldn't. The rest of the members should be fairly self explanatory.
Then to see this in action, it's a matter of opening the website in iOS Safari and clicking the 'Share' button, then clicking 'Add to Home Screen'. Then clicking 'Add'.
A new "App" icon will appear which will have the name of the website as specified in the manifest file, and tapping that app will open the website in a new screen.
This does not remove the need for creating a dedicated app, nor does it remove the need for making a responsive website for mobile devices, but it is a nice little extra to offer your users.