I think there's very likely a connection between the two, but not a direct one.
The first difficulty will be visible when you dig into the literature on Dual Process Theory — that's the overarching theory that describes what is commonly referred to as System 1/System 2. The long list of descriptors in the table on that Wikipedia page will hint at the problem: variations on the theory have been used to explain so much that it has been difficult for researchers to agree on a neurological basis.
Two good academic articles:
Keith Stanovich and Richard West coined the "System 1/System 2" term in a 2000 paper: Individual difference in reasoning: implications for the rationality debate? in Behavioral and Brain Sciences, which was accompanied by a broad response of peer comments.
A dozen years later, Jonathan Evans and Stanovich wrote Dual-Process Theories of Higher Cognition: Advancing the Debate. There are undoubtedly important papers which are more recent, but I think what you'll discover is that the theory, while useful, is controversial.
The connection that bridges the two probably resides in Executive Function. Just to pull a paragraph from a paper that illustrates this:
The dual-process theory suggests that less efficient dopamine-mediated
reinforcement processes and deficient extinction of previously
reinforced behavior may explain behavioral changes often described as
poor executive functions or as response disinhibition.
That invocation of a dual-process theory probably won't sound familiar to anyone whose exposure to the theory was by way of Kahneman's book, but you can see that the connection is via executive functions.
If you amend your search to connections between Executive Function and ADHD, you'll hit pay dirt quite quickly. A good article to start with is Thomas Brown's ADD/ADHD and Impaired Executive Function in Clinical Practice (2008).
And it becomes fairly clear that there probably is a connection: ADHD probably interferes with either or both of System 1 or System 2. But which, and how?
Getting to your explicit question: Do individuals with ADHD rely more on System 1 thinking? Since System 1 thinking is, broadly speaking, subconscious, the answer is probably not. Maybe a little. It seems more likely (to me) that ADHD shifts the focus of thinking in ways that, perhaps, disconnects some System 1 processing from the System 2 thinking which we might expect to have followed it, but that doesn't mean that less System 2 thinking is taking place. A very different hypothesis could assert that some System 1 process is degraded, leading to a larger amount of poorly directed System 2 thinking. There doesn't seem to be much research targeting this question yet.
With respect to the Test of Variables of Attention, note this text from the "About" page:
The duration of testing is a significant factor, since subjects who
are older and more intelligent can compensate for mild or moderate
attention problems for 5, 10, even 15 minutes.
That word "compensate" should give you pause. Is it plausible that some automatic System 1 response is detecting attentional drift and correcting it? Or that, with age, some subjects will have compensated by consciously directing (System 2) themselves to refocus?
The 2013 blog post you link to is interesting, but appears to be a little haphazard. The blogger seems to believe that a fast or "global thinker" could trigger a false-positive for ADHD. I'd suggest looking into the research connecting ADHD and high IQ. There's not too much of it, but it is interesting. Two articles you might start with: Executive Function Impairments in High IQ Adults With ADHD (2009) (although the focus is on adults, not children) or Co-Occurrence of ADHD and High IQ (2010).
