Fibonacci Sequence, Beginning with a ‘New’ Number?

I love using and exploring the Fibonacci sequence in my work, but something about it has always bothered me. Traditionally, the sequence begins as follows:

0, 1, 1, 2, 3, 5, 8, 13,…

What is strange to me are the way the sequence begins and how the first two numbers, 0 and 1, break the sequence's own defining rule—that each term should be the sum of the two preceding terms. When you look it up, you often see explanations like “it has to start somewhere.” This answer doesn’t feel entirely satisfying. If the sequence is truly based on self-propagation, shouldn’t it begin with a value that inherently satisfies its self-referential nature?

There’s an obvious truth that the meaning of the sequence is “correct” as it carries weight in the natural world. People have explored this at length — from sea shells to human bodies, the Fibonacci sequence exists materially everywhere in nature. However, to this point, shouldn’t the beginning of the sequence satisfy the rules of the sequence itself? If anything, it’s imperative for the Fibonacci sequence in particular because part of our interest in the sequence is it’s fractal nature.

Look, I get it. The sequence required the jumping off point to even exist in our reality until this point. However, maybe we can explore the Fibonacci sequence in a way that uses paradox — if anything, to me, having the sequence begin with a paradoxical “number” or symbol might unlock some deeper truths about the birth of our physical reality.