Are the Alpha and Theta brainwave states showing something always ongoing, but only experienced by the brain when the conscious thought is relaxed and, in a sense, goes looking for that particular level of consciousness?

This is a perceptive and deep analysis you have come to about the workings of the mind as probed by scientific instrumentation. This all follows logically from all you know and understand about the workings of the mind, that it is only the conscious mind that sleeps. The levels of the mind which are deeper are always thinking one thing or another and always available to communicate with the conscious self if that is possible, the exception being the deep subconscious that is cut off completely from conscious awareness. There can only be crosstalk at certain times, in certain ways; for example, during the dream state where the deep subconscious might visit the upper subconscious while dreaming and impinge some thoughts or impressions. There is also a certain degree of intuitive interplay that may also be perceived intuitively by the upper subconscious, for example with a recognition of things found interesting by the deep subconscious, or worrisome. So the true nature of what is measured clinically with the EEG during maneuvers to go into a meditative state or hypnotic trance, for example, or while falling asleep, are reflecting what the conscious level of the mind is connecting to and focused on, and that will, of course, only show up when the conscious mind lets those other parts of the mind have a stage within the brain to interact. So the conscious self truly must stand down or stand aside to a significant degree and let the subconscious or cellular memory chime in with an impression of something. That will float up as an internal thought, just like all other internal thoughts, because they are operationally identical in how consciousness is perceived by the brain. But this insight has value in thinking about what science might say about your ideas and how to help better explain the true workings of things.