Depends on your definition of "works".
This is, as usual, very simple and practical.
See, if my router dies, then I am off the Internet - and that's a problem. So I should have some assortment of redundancy and/or spare parts to harbour no single-point-of-failure. So far, I suppose, most of us know that.
But then, if my desktop dies --- I still have Internet, only I cannot browse it with a headless router. So, this is also a single-point-of-failure. (Okay, if you have some working laptop or tablet, then you're out of that dilemma. I currently don't, and after finally moving all to amd64, I am a bit low on spare parts. I would actually need to start installing some X on the router, move the screen there, and work it while standing in the hallway - thats about as bad as using the smartphone for browsing.)
A "modern" browser is a full-featured portable application platform, with HTML/CSS as UI description languages. Sure, something like this can't ever be "light-weight"
This is again very simple and practical. What I need to do in the mentioned case is, browse some merchants and/or ebay for the potentially needed replacement parts. And go thru the process of electronically placing and paying an order.
I very much doubt this would still be possible without a "modern" browser, specifically javascript.