The psychology school once argued that the architect of the controllable "Skinner box" and programmed learning. Their key tent is that observable behaviour may be scientifically studied. So, we ask, if the idea is justifiable.
Only observable behaviors can be recorded, documented and analysed. If the behavior is as a result of an intention then it can be recorded but one would never know the intention of the person, one can only guess. Speech is a behavior that can be observed including the content, inflection and nuance as well as emotions expressed. What goes on in someone's mind cannot be observed except perhaps via in situ brain scans with electrodes connected to various parts of the brain.
Behaviorism is a beautiful Psychology theory with the scientific virtue of parsimony - the most simple explanation to explain as much as possible. Treat an organisms as a "black box" and explain all behavior with stimulus and response. In its historical context, I feel Watson's and Skinner's worldview was justified given the failure of early schools of Psychology - notably the failure of introspection for Structuralism. Other contemporaries of Behaviorism were making progress with models referencing mental processes (e.g., Development by Piaget, Cognitive by Ebbinghaus), but none were so revolutionary at the time at producing practical applications.
I feel it's no longer tenable to explain all Psychological phenomena without reference to internal states. Behaviorism strains too much trying to explain: language development, instinctual drift, cognitive dissonance, temperament, observational learning, latent learning of mental maps, etc. Still, Behaviorism remains a very effective practical model for many real world applications (e.g., ABA for treatment of children with Autism Spectrum Disorder).
You ask: " Why should behaviour such as - speech, intentions and mental processes [be] disregarded as unobservable in research?" In AT LEAST a significant indirect sense the answer is THEY SHOULDN'T ! Every concept should have clearly-relevant directly observable KEY overt counter-parts at least some time (point) in ontogeny (aka child development). I believe qualitative changes in thought and thought processes have subtle, but overt directly-observable behaviors patterns AT THE INCEPTION OF EACH stage/level of such thought (eye tracking technology, etc. will likely be necessary to capture this; it is what I see as perceptual shifts and then as perceptual/attentional shifts) . (We are thought to have 5 such stages/levels, with several theorist/researchers saying they have 2 phases, each,)
I think the way of dealing with things related to some intentions and to thought processes is to find the early directly observable overt behaviors AT THE INCEPTION OF EACH NEW QUALITATIVE LEVEL OF CONCEPTUAL THOUGHT: I have outlined how this could be done. See:
AND ALSO SEE the following to get an idea of wrongly self-limiting sorts of thought: https://www.researchgate.net/post/Do_we_lack_imagination_for_how_great_our_imagination_representation_of_things_AND_ACROSS_CIRCUMSTANCES_SITUATIONS_can_be
AND see my assessment of the current deficiencies of current ideas of "abstract concepts" -- at least as limits here are countered by my perspective; specifically see my assessment of the following, in my Comment under it (WITH an alternative to All THAT offered):
Why should behaviour such as - speech, intentions and mental processes [be] disregarded as unobservable in research?
You are right in that these cognitive processes are largely ignored in research, but probably understandably so as a mental process such as an intention is hard to measure – but in theory it is not impossible to measure.
And while the greater emphasis on various types of brain scanning procedures in neuropsychology has not really progressed our understanding of the where and how of cognitive processing, I think greater strides can be made in these areas if brain scanning techniques are coupled with a greater emphasis on studies using more experimental designs – specifically using within participant as well as between participants designs.