Your last paragraph gets it right: the Hamiltonians are different, and they're not always available.
The Unruh effect is a prediction of Quantum Field Theory in Curved Spacetime, so I'll first discuss how one formulates QFT in general spacetimes before focusing on the Minkowski case.
In flat spacetimes, we typically understand states in terms of particles due to Poincaré invariance. The fact that Minkowski spacetime is Poincaré invariant lets us single out a quantum state that is Poincaré invariant, which we call the vacuum. We can then interpret other states as excitations of this vacuum, and hence understand them as particle states. This is the short version. Notice this is always done in an inertial frame of reference, and (orthochronous) Poincaré transformations will keep this vacuum invariant (i.e., all inertial observers agree on what is the vacuum).
Consider now a generic spacetime. Poincaré invariance now means nothing: general spacetimes lack Poincaré symmetry. Hence, there is no way of defining a preferred state. One can no longer choose a vacuum. And that's it. Without a vacuum, you have no way of defining a notion of particle. This is not a problem, because QFT is a theory about fields. Particles are just a convenience we can use on some situations.
However, suppose the particular case in which you have a timelike Killing vector field. In other words, you have a generalized notion of time-translation symmetry. In this situation, you can (roughly speaking) understand what would be the Hamiltonian for the theory as seen by the observers whose trajectories follow that Killing field. Using this Hamiltonian, one can then once again speak of a vacuum that minimizes energy. Notice that this will lead you to a time-independent Hamiltonian (with respect to the "time" defined by the Killing field).
What happens in the Unruh effect is that you have two possibilities in Minkowski spacetime. Usually, we pick the Poincaré time-translation as a Killing field. However, another option is to pick the Killing field associated with boosts along some Cartesian direction (say $x$), and restrict it to the region $x > |t|$ of the spacetime (known as the right Rindler wedge). In this region of spacetime (which can be understood as a spacetime of its own right), we can then define a vacuum by minimizing the Hamiltonian defined in "boost time", and get a different notion of particle.
It then remains to understand what this means. It turns out that the observers that follow the boost Killing field are accelerated observers, so we can interpret this result as telling us how accelerated observers define the notion of particle, which disagrees with the definition of inertial observers. If we leave it like this, it would definitely sound sketchy. However, another way of studying these themes is by using particle detectors (couple a 2-level system to the field, so it can work as a quantum system that tells you when a particle is seen). The accelerating and inertial particle detectors will then react differently to the field, in perfect agreement with the prediction that the accelerated detector should be interacting with a thermal state.
It might also be worth pointing out that, while there is no direct experimental evidence for the Unruh effect, I should mention two papers:
- arXiv: gr-qc/0205078 "illustrate[s] how the Fulling-Davies-Unruh effect is indeed mandatory to maintain the consistency of standard Quantum Field Theory" by studying the decay of accelerated protons (while inertial protons don't have enough energy to decay, accelerated protons have an external source of energy)
- arXiv: 1701.03446 proposes an experimental procedure to measure the Unruh effect and predicts its result with standard classical electrodynamics by relating it to radiation emitted by accelerated charges. It mentions that "Unless one is willing to question the validity of classical electrodynamics, this must be seen as a virtual observation of the Unruh effect".
For a more technical post on how to define QFT on curved spacetimes, you might be interested in this answer I wrote a while ago. For a more "SciComm" view (still a bit technical though), there's also this answer, also by me. For more information on these topics, I recommend Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. For more on the particle detector view, you might be interested in the paper by Unruh, W. G. and Wald, R. M. (1984), "What happens when an accelerating observer detects a Rindler particle", Phys. Rev. D 29, 1047.