You seem to be exaggerating the generality of the causal Markov condition (CMC) when you say it is deeper and more general than the second law of thermodynamics. In a big world, failures of the CMC abound. Let's say the correlation between the psychic cousin's predictions and the top card of the deck is explained by the person performing the test being a stooge, who is giving some non-verbal indication to the purported psychic about the top card. So here we have a causal explanation of the correlation, as the CMC would lead us to expect. But since we are in a big world, there are a massive number of Boltzmann brains out there, outside our light cone, whose brain states correlate with the top card in the same way that the cousin's does. But there is no causal explanation for this correlation, it's just the kind of thing one would expect to happen, even non-causally, in a sufficiently large world. So the CMC isn't a universal truth.

Now, the CMC is a remarkably accurate rule if we restrict it to our local environment. But it's pretty plausible that this is just because our local environment is monotonically entropy-increasing towards the future and entropy-decreasing towards the past. Because of this feature of our environment, local interventions produce correlations that propagate out spatially towards the future, but not towards the past. When you drop a rock into a pond, waves originate at the point the rock hit the water and travel outwards towards the future, eventually producing spatially distant correlations (like fish at either end of the pond being disturbed from their slumber).

Imagine that there is a patch somewhere in the trackless immensity of spacetime that looks exactly like our local environment, but time-reversed. Here we would have a pond with a rock initially lying at its bottom. Spontaneously, the edges of the pond fluctuate so as to produce a coherent inward-directed wave, which closes in on the rock, transferring to it sufficient energy to make it shoot out of the pond. If you don't allow backward causation, then it seems that the initial correlated fluctuation that produced the coherent wave has no causal explanation, a violation of the CMC.

The second law is often read as a claim about the condition of the early universe (or some patch of the universe), specifically that there were no correlations between different degrees of freedom (such as the positions and velocities of particles) except for those imposed by the macroscopic state. There were no sneaky microscopic correlations that could later produce macroscopic consequences (see this paper). Entropy increase follows from that, the story goes, and, plausibly, the success of the CMC follows from that as well. There is a strong case to be made that the second law is prior to the CMC in the order of explanation.