For sure presentations aimed at lay audiences tend to oversimplify and exaggerate research findings, but for the underlying construct, there is a great deal of evidence, and it is considered widely accepted in the field.
The technical term for it is dual-process theory, and is often referred to as System 1 and System 2 based on the popular book Thinking, Fast and Slow, by Nobel laureate Daniel Kahneman, which is a very accessible source for anyone interested in the evidence-base behind this theory (or class of theories, as many have been proposed). It is also a popular topic on this forum, so be sure to search for these keywords to learn more (examples: Is Decision-Making Emotionally Based, with Rationalization as the only Conscious Component?, Does fixing cognitive biases do more harm than use?, Research on aversion to cognitive effort?).
One quibble with the way some non-technical presentations refer to this construct is that they seem to imply that System 1 and 2 refer to distinct physical "parts" of a person. For example, the left brain / right brain division can be a reasonable metaphor for lay persons, but it should not imply that a stroke in one half of your brain will turn you into Spock (an emotionless perfectly logical decision-maker). It's important to emphasize the process part of dual-process theory, and to this end, authors in the field often use the terms Type 1 and Type 2 thinking instead. For example, Type 1 (intuitive) thinking is likely to be composed of many sub-processes working in parallel, and Type 2 (rational) thinking surely depends on these processes for its own contributions.
Consequently, asserting a good/bad dichotomy to these processes is misleading, as they serve interdependent purposes. There are many scenarios where Type 2 thinking is impractical due to time constraints and bounded rationality. Evans and Stanovich, two of the leading researchers in the field, agree (excerpt from Evans, 2012):
In recent writing, I have attributed responsibility for a range of
cognitive biases roughly equally between type 1 and type 2 processing
... Stanovich ... similarly identifies a number of reasons for error
other than a failure to intervene with type 2 reasoning; for example,
people may reason in a quick and sloppy (but type 2) manner or lack
the necessary “mindware” for successful reasoning.