I would say both. User in the sense that people expect the web to behave and be just as rich and fluid as a native app (and will often leave or mistrust you otherwise) and Client in the sense that having a rich web experience drives revenue and whatever other business metrics.

> it is staggering how easy it is to break many of those sites just by resizing the browser window.

I should say that a responsive website does not actually mean a website that responds to resizing the browser window - and that should not be taken to mean it's "broken". Saying that makes you sound like a typical sweatshop PM or stakeholder :) - i won't belabour the actual meaning here.