Most psychologists adopt the basic theory of our culture, which includes that thoughts are symbols with no hidden meaning. According to that theory, felt senses should not exist.
They do have trouble explaining their data. One famous psychologist, Gordon Bower, revisted one of the basic phenomenon in psychology. Subjects first learned associations such as dog-table. This means that when the experimenter says "dog", the subject is supposed to say "table". Then they learned a second set of associations which had competing responses. So, to continue this example, dog-piano might have been in the second set.
The standard finding, which Bower replicated, is that learning the second set interferes with "memory" of the first set. In other words, given dog and asked to give the response to the first set, which is table, they were less likely to be correct.
The standard explanation is that the two associations compete. In the thoughts-as-symbols model, this means that the subjects presumably said "piano" when they should have said "table". Bower however let his subjects guess as many times as they wanted. He found that learning of dog-table was still impaired. Not only did it not appear first, like it was supposed to, it just disappeared from the listing.
At the time he did this study, Bower had no explanation. But it is easily explained with the logic of felt senses. The stimulus "dog" now produced a complicated thought that included both table and piano. It was difficult to extract the information "table" from this thought.
Parallel Distributed Processing
There is another branch of cognitive psychology that tries to describe thoughts as if they were a collection of neurons either on or off. This simplifies but mimics what probably happens in that brain. These models are never used to predict the existence of felt senses, but they almost necessarily must.
For example, in the Bower experiment, the subjects are simply building an association such that "dog" produces "table" and then another association such that "dog" produces "piano". Now when they are given dog, with modest forgetting of the second association, you have a "brain" that produces both.
These types of models are often used to exlain the formation of concepts. Essentially, if you add together similar thoughts, you end up with a concept that contains the similarities in the thoughts and blurs out the differences.
Again, all you have to do is combine together dissimilar thoughts and you have felt senses.
I have not looked at recent psychology, but as of 10 years ago, there was a disconnect. Felts senses come up as a concept in psychotherapy, so cognitive psychologists would not hear of them. So they do not try to explain them. Focusing people, on the other hand, have no reason to study study the complicated models which unknowingly would produce felt senses. Experimental psychologists, stuck in the middle, are accustomed to findings which they cannot explain.