For sufficiently large intervals, this is true; it follows from Buchstab's theorem (see Montgomery and Vaughan chapter 7.2) that, for sufficiently large $N$, there are approximately

$$ e^{-\gamma}\frac{2p_n}{\log p_n} $$

numbers in the interval $[N,N+2p_n]$ with all prime divisors at least $p_n$. For all primes at least 7, this is larger than 3. The cases 2,3 and 5 can be checked directly, as you mention in your question.

I imagine that if you were interested, it should be fairly easy to work out a specific (albeit large) lower bound for something like the above to hold, and then run a computer search on all smaller intervals.

Of course, there may well be a simple elementary way anyway, which I can't see right now.