Questions regarding SETH
The subtlety comes in where we introduce the notion of "harder". The reduction showing that SAT can be reduced to Hamiltonian Cycle shows that the latter is "harder" up to polynomial factors. In doing the reduction, we may very well increase the variable $n$ in question substantially (by some polynomial). And the canonical reductions do blow it up quite substantially.

So what's the upshot? Suppose that the reduction from a SAT instance with $t$ variables yields a HamCycle instance with $n = p(t)$ vertices for some polynomial $p$. Running some $O(2^{\epsilon n})$ time algorithm on this HamCycle instance takes time $O(2^{\epsilon p(t)})$. If $p(t) \ll t/\epsilon$, you just broke SETH. But in general, $p$ may be larger than that (often not even linear), in which case your reduction takes longer than the naive brute force approach.

Some recent papers actually work asking the same line of reasoning that you do to establish SETH hardness. For example, consider the graph diameter problem, where we are asked to determine how far apart the two furthest nodes on a graph lie (that is, we're asked to compute the maximum over all (u,v) pairs the length of the shortest path between u and v). Roditty and Williams showed that even in the unweighted case, where all edge lengths are 1, the problem is difficult to solve in time $O(n^{1+\epsilon})$ (using your notation with $\epsilon&lt;1$). The way they do this, roughly, is by finding an exponential-time reduction from SAT to GraphDiameter that blows up the number of vertices from $t$ to $O(2^{t/2})$. In this new giant instance, the diameter is 3 if the instance is satisfiable, 2 if not. Thus, if they can solve GraphDiameter in time $O(n^{1+\epsilon})$ (and if the original exponential time reduction is fast enough, which it is), then they can solve SAT with $t$ variables in time $O((2^{t/2})^{1+\epsilon}) = O(2^{\frac{1+\epsilon}{2}t}) = O(2^{\epsilon't})$ for $\epsilon'&lt;1$, breaking SETH. In reality, their reduction works through dominating set first, but the key ideas remain the same.