You hear these examples a lot. But I think if you look closer you see that these are examples stories being more complicated than their stripped down versions, not counterexamples. The stripped down version still holds to the pattern.

The "real" beginning of Starbucks was an entrepreneur buying Starbucks and merging it with another couple of coffee shops then expanding immediately at serious pace. It went public 5 years after the purchase.

Mcdonalds is a similar story. Bought and expanded.

It's hard to measure the exact beginning of these businesses, they weren't founded with the intention of raising capital and going big. They did however join that trajectory at some point. The interesting thing is that they joined it very close to the launch point - these weren't small companies that slowly became medium then big, these were tiny companies that stayed small for decades, then they became giants very quickly, almost instantly.

edit: BTW, it is definitely possible of finding companies that grew slowly over decades. But in those cases you probably won't find that they were small businesses for a long time. They spent roughly equal amounts of time as 5 person companies, 10, 100, 1,000 - so you don't here so much about their heritage as a small business. They might have only spent a few years that way.