Bear in mind, in the US you (normally) just apply to a university "at large". You don't apply to a specific subject/degree programme (with a few exceptions, increasingly engineering departments have you pre-apply to the major), as you choose your major sometime before then end of your sophmore year typically. This means the university admitting you needs to be reasonably confident you can do pretty well in any major, hence the well-roundedness "requirement".
Related to this typically US colleges (with one or two exceptions, like Brown) have a structured "core" of general education requirements, where all students are expected to take one or two classes (i.e. modules) in different areas (e.g. maths/quantitative skills, sciences, humanities, social sciences, arts, writing, foreign languages) - this supports students in finding which major to go into, and necessarily requires at least some "well roundedness" for any student, even the ones who already know (or think they know) what they want to study. You can't just apply to university in the US to study e.g. physics - you inevitably also need to take a bunch of other classes. This has it's upsides and downsides but that's another debate entirely...
The UK system somewhat engenders a more meritocratic format because you're applying to an individual subject area, so they can just focus on those relevant aspects, and the narrower focus allows admissions tutors in the UK to use more common criteria to compare candidates (e.g. A-level UMS). This is also supported by the centralised application format through UCAS, so everyone has their information submitted to all universities in more or less the same format. In the US, you normally apply to the individual universities, unless they're part of the CommonApp (which is kind of like UCAS, but even then the essays i.e. personal statements are still sometimes unique to different colleges) or a state school system like the UCs.