You're incorrect that HMC is not a Markov Chain method. Per Wikipedia:
In mathematics and physics, the hybrid Monte Carlo algorithm, also known as Hamiltonian Monte Carlo, is a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. This sequence can be used to approximate the distribution (i.e., to generate a histogram), or to compute an integral (such as an expected value).
For more clarity, read the arXiv paper by Betancourt, which mentions the NUTS terminating criteria:
... identify when a trajectory is long enough to yield sufficient exploration of the neighborhood around the current energy level set. In particular, we want to avoid both integrating too short, in which case we would not take full advantage of the Hamiltonian trajectories, and integrating too long, in which case we waste precious computational resources on exploration that yields only diminishing returns.
Appendix A.3 talks about something like the trajectory doubling you mention:
We can also expand faster by doubling the length of the trajectory at every iteration, yielding a sampled trajectory t ∼ T(t | z) = U T2L with the corresponding sampled state z′ ∼ T(z′ | t). In this case both the old and new trajectory components at every iteration are equivalent to the leaves of perfect, ordered binary trees (Figure 37). This allows us to build the new trajectory components recursively, propagating a sample at each step in the recursion...
and expands on this in A.4, where it talks about a dynamic implementation (section A.3 talks about a static implementation):
Fortunately, the efficient static schemes discussed in Section A.3 can be iterated to achieve a dynamic implementation once we have chosen a criterion for determining when a trajectory has grown long enough to satisfactory explore the corresponding energy level set.
I think the key is that it doesn't do jumps that double, it calculates its next jump using a technique that doubles the proposed jump's length until a criteria is met. At least that's how I understand the paper so far.