"Almost all objects have property P" vs. "It is easy to test whether an object has property P"
They are separate (assuming $P \ne NP$).  Consider the following property $P(x)$: $x$ is a $2n$-bit string, where either the first $n$ bits are not all zeros, or the last $n$ bits are a yes-instance of 3SAT.  It's clear that testing whether $x$ satisfies $P$ is NP-hard, yet almost all strings satisfy it: the density $\to 1$ as $n \to \infty$.