Why not support an interface on something that's not a phone too? It doesn't sound like supporting a web SPA would be stretching their core competence.
They have an emergency-access version at http://web.monzo.com but they're a mobile app bank.. not really much more than that to say, website wanters aren't the market they're serving
What's the big difference between website wanters and app wanters? Don't most people want to sit down on a chair at a keyboard and large screen if possible?
The UI of the monzo app is actually pretty good on a small screen. I've never felt I need to use a desktop to view it (same for the "traditional bank" app I use too to be honest - going desktop doesn't really add anything since its just a table of numbers really).
The reality is that you never really use the app anyway apart from very occasionally. The "UI" for this bank account is the card itself.
LOL, whilst I won't necessarily include all Android devices in this, I'd argue that iPhones and better Android setups are likely the Most secure device that most people own.
Traditional PC security is way too complex for non-technical people to manage effectively, so the approach of walled gardens and vendor managed security is the way to go.
Not to say that a technical person can't achieve a better level of security with a PC/Mac, but for non-technical users, it's going to be mobile all the way.
It being an app, downloaded from a moderated app store (albeit not a very well moderated one) rules out an entire class of attacks (phishing), which make up a large portion of online banking attacks.
This alone is probably a decent competitive edge over traditional banks.