I think the words we use to think about how the mind works help limit our understanding.
For example, we refer that part of though, or intuition, whatever, that we don't do consciously (what does that even mean) as occurring subconsciously. If I understand correctly when people say 'subconsciously' they are refer to what happens outside of our awareness, as in like you say we don't sit there with a dictionary or a bunch of idea reference books and flick between the pages in order to form a thought.
But why the term subconscious.
sub-
1. a prefix occurring originally in loanwords from Latin ( subject; subtract; subvert; subsidy); on this model, freely attached to elements of any origin and used with the meaning “under,” “below,” “beneath” ( subalpine; substratum), “slightly,” “imperfectly,” “nearly” ( subcolumnar; subtropical), “secondary,” “subordinate” ( subcommittee; subplot).
I conjecture that using the word subconscious causes a linguistic trap that results in us undervaluing what goes on in our minds.
Also I think what most people think of when they think of thinking is merely the subvocalisation[1] or internal dialogue we run in mind. Here I agree with what you've written that, actually all, logical connections are done with intuitive reasoning.
NB I'm a complete armchair amateur at philosophy / psychology / neurology. Of course everyone's got a consciousness so we're all experts. Right.
1. https://en.wikipedia.org/wiki/Subvocalization