That's a rather confused account of the matter.

The rebus principle where someone might use a depiction of an eye for the sound "I" and so forth is the very basis of the script and was there from the beginning. The complicated part is they'd use words with one to three consonants and strip the vowels. To continue the example, we might use 𓃠 to represent the consonants "ct" and thus use it to write "cat", "cot", and "cut."

There was an inventory of uniconsonantal or uniliteral signs dating back to the very beginning of the language which the ancient Egyptians could have used as an alphabet (or abjad if we want to be pedantic) if they had wanted to, but they never did—at least to write Egyptian. The basis of our alphabet, Proto-Sinaitic script, seems to have come about when speakers of Caananite languages in the Sinai Peninsula borrowed a small number of Egyptian hieroglyphs, assigned them the phonetic value for the thing depicted in their own Caananite language, and they didn't bother with anything other than uniconsonantal signs.

The "three different alphabets" thing is unrelated to any of this. Hieroglyphs and hieratic appear around the same time. Hieroglyphs were used for monuments and more formal contexts. Hieratic is a cursive form of hieroglyphs that was much faster to write with a brush pen and ink. It tended to be used for literature, correspondence, and record-keeping. From what we know of Egyptian scribal education, they started out with hieratic and then moved on to hieroglyphs, with not everyone progressing to the point where they started learning hieroglyphs. This is quite the reversal from how we approach things today, with virtually every student of ancient Egyptian language learning hieroglyphs (specifically, Middle Egyptian) first and then moving on to learning hieratic. Demotic was a later evolution of hieratic. And eventually, the Egyptians wrote their language using a modified Greek alphabet ultimately derived from their hieroglyphs (Coptic).