You rightly say that “Intuition is all about the first gut feeling.” But also note that we “may unconsciously dismiss ideas…”

This is the key - that most of us live UNconsciously, unaware of our feelings, ideas, thoughts, moods, etc. Instead we are identified with them and thus there is no “watcher” of them.

You suggest: “Watch out for when you’re trying to suppress the first idea that comes to mind…” But most people cannot act upon this suggestion. An idea that comes to mind either quickly disappears without being noticed (or “watched”) or takes over the mind completely (identification) - again no “watcher” is active to notice the idea.

Throughout millennia wise teachers have exhorted us to live consciously and suggested various ways to activate the conscious “watcher” in us. Like a muscle, the watcher gains strength through repeated use - or practice. The deliberate (conscious) activation and strengthening of the “watcher” in us goes under various labels:

Watch, Know thyself, Self-awareness, Mindfulness, Self-observation, Witnessing, Inward-looking, Self-remembering, Meditation …they all mean the same. And all require an OBJECT to be witnessed/watched/observed. That object is one of the five senses.

One could “watch” the touch of breath at the nose (sense of touch). This would gradually strengthen the ‘watcher”. Alternatively, one could “watch” the touch of light on the eyes (sense of sight). This is achieved by looking slightly upward or by simply noticing one’s peripheral vision. Here are a couple of portrayals of this practice from around the 1460’s…

http://www.artrenewal.org/asp/database/image.asp?id=21815
http://www.artrenewal.org/asp/database/image.asp?id=21816

…Over many months of practice the “watcher” is strengthened and intuitive ideas are then actually noticed for what they are - appropriate suggestions deserving consideration for action.