Lack of rigour in Spivak's Calculus book?

I think we can reasonably state that an ellipse is both $1)$ existent and $2)$ not a straight line.

The reality is that, if we are $100\%$ rigorous in everything we say, any textbook would be thousands of pages long. Each statement would have to be proven from axioms, and nobody would want that. There are certain reasonable things we can and must assume.

I also will say that I understand the frustration. Sometimes textbooks are frustratingly casual where you don't want them to be. But part of writing math well involves knowing which things are okay to exclude.

Looking at the PM will probably be a good experience.

As for "rigor/transparency", as others have mentioned every single line of written math will have implicit assumptions, and these depend on context. When Spivak requests that $1\ne0$ he is discussing the axioms for the real numbers; eventually, the properties of the reals are taken for granted (or do you expect axioms every time he goes from, say, $5x+1=0$ to $x=-1/5$?).

In your concrete case of the ellipse, for each point $(x,y) $ you have a triangle where one side is $2c $ and the sum of the other two is $2a $. This immediately implies that $c <a $. Spivak's reasoning starts from the intuitive idea that you are defining the ellipse in the real plane, so it is sound.

Besides, "we must clearly choose $a>c$" refers to the formula, where $a^2−c^2$ appears as a denominator and so it has to be nonzero. So it is clear, indeed.

I agree with you 100%: Spivak's definition is sloppy. He says:

A close relative of the circle is the ellipse. This is defined as the set of points, the sum of whose distances from two "focus" points is a constant.

By this definition, the line segment between the two foci is an ellipse. When he says later, "we must clearly choose $a > c$", he is contradicting his own definition.