In lieu of other answers: in the U.S., essentially no undergrad program allows focusing on a single subject. The UK schools are the opposite, essentially.
To ask whether we should "allow/encourage" a student to focus on only one thing...?
And, yes, there's the pragmatic issue of whether parental or other essentially not-professional-research-mathematician opinions are accurate. E.g., not every tall person has a future in the NBA.
In fact, Hahvahd, Princeton, and such [EDIT: I was unaware until I saw @Zev C's answer that Brown allows essentially unlimited flexibility in course selection. Pretty amazing! Unusual among research universities. The "self-designed major" is more typically possible only at small colleges in the U.S.] ... do require considerable exertion outside one's field of special interest. For a kid with a rather narrow capacity-of-interest, a big state school (e.g., Big-Ten, ...) might actually be better: the out-of-subject classes are not super-strenuous, and the "graduate" courses in math will probably be constructive, but not toooo critical. (We anticipate that, even if a kid has some gifts/talent, it doesn't always quickly mature to allow writing coherent, adult English (or whatever) narratives concerning serious issues.)
These days, faculty at at-worst-second-tier universities are mostly graduates of top-tier universities, so understand the sensibilities of those places, and, for that matter, know the people at the top places, and can communicate on behalf of a kid who deserves a place there...
My bonus comment is that there is considerable fragility in choice-to-exclusion-of-other... when a teen decides math-is-it. Much as a kid in an "arts high school" blows-off all other subjects. It is a gamble. We may be sympathetic to the gamble, but, as responsible adults/advisors, we should hedge the bets of our proteges, on their behalf, no matter their "enthusiasms".
In my own life, by accident, I went to a state school (Purdue), and was given great latitude... understood in hindsight... "to figure things out" mathematically. The environment was perhaps a little too "quiet", but for a young kid this may be extremely useful. That is, over-stimulation is not a positive thing for many (capable) people. Even when one eventually discovers that people had thought about one's current "pet" issues 150 years ago, it is xxxtremely helpful to have had a quiet, safe environment to have developed the enthusiasm for the (long dead, but, hey, interesting...) issue.
From another tack: some of my fellow grad students at Princeton long ago were from much fancier places than Purdue, ... and/but they didn't retain the affection for mathematics that the lower-stress environment had allowed me.
So... A personal interview is absolutely necessary, to figure out what a kid needs. Prestigious undergrad institutions are all very nice... if one can distinguish oneself... but one's career is initially determined by the grad school. Sooo...
(That's all I know... :)
