It may be helpful to provide a combinatoric perspective for contrast. It will raise some interesting questions.

Given $n$ equal to the $k$th prime represent the left hand side as a reduced fraction I call $Pk$. For $k=1$ to $4$ we have the fractions 1/2, 1/3, 4/15, and 8/35. Thus I would expect an interval containing numbers coprime to $210$ to have this density be 8/35 only if the length of the interval is a multiple of 35. This quickly leads to the question: how fast does the denominator of $Pk$ grow?

A related statistic is the ratio $\omega(\prod (p-1))$, the number of distinct prime factors of the (unreduced) numerator of $Pk$; for $k\lt 46$ this quantity is lower bounded by $k/3$. I do not know the asymptotic of this quantity, but to discuss the previous paragraph further let me conjecture that it stays above $k/3$.

If it does stay above $k/3$, this means that the reduced denominator of $Pk$ is (very roughly) at most $P^{2/3}$, where $P$ is the product of the first $k$ primes. So for such $P$ (and thus for $X$ not far from $P$), I would expect an interval of length about $P^{2/3}$ to exactly match the density of coprimes to P in the interval $(\log P, P) \approx (p_k,P)$. If you are willing to accept some error, I would expect much smaller intervals to exist (of almost any length greater than say $2\log P$) to come close to the desired density. ( Of course, I am interested in how much the density can vary at that scale, which I believe can go from 0 to almost twice the average.) As remarked in another answer, if you fix $\theta \gt 0$ and ask ( as $X$ goes to infinity ) about density of numbers less than $X$ with no prime factor less than $X^\theta$, that density drops considerably.

Gerhard "It's A Matter Of Scale" Paseman, 2016.12.06.