If the long-term values of a successful education in mathematics are general skills like: analytical and logical thinking, ability to abstract and see patterns, modelling and problem solving; well then perhaps it does not matter that much what parts of mathematics are studied? Why algebra?
I think algebra is special for several reasons. It is the next step beyond elementary arithmetic. Algebra is really where mathematics break loose from everyday practical concerns of measuring, counting and reckoning. Historically it also marks the beginning of European mathematics going beyond the classics with Vieté systematically introducing letters in equation-solving, continuing with Descartes and Fermat introducing letters in the geometry of curves (of course the real history is much more complex).
Algebra is also necessary for anything but the most primitive function theory and consequently for the calculus. As far as I understand all of mathematics is now couched in algebraic language, even geometry.
So teaching mathematics without using algebraic notation and concepts seems to be a contradiction in terms unless we restrict ourselves to verbal mathematics and Euclid style synthetic geometry and projective geometry.
Even subjects proposed as replacements such as “quantitative literacy”, statistics and logical reasoning would be hard to carry very far without some kind of algebraic language.
It now seems to me that algebra (or algebras), with their syntax, semantics and pragmatics are really the languages of all parts of mathematics. When first encountering algebra you have to grasp the idea of a symbol (rooted in a set of “values” for instance the real numbers) standing in for an object carrying with it a set of operations that can be performed on it. Obviously – as all teachers experience – this is hard for almost everyone to wrap their heads around.
In algebra you can learn about modelling and problem solving: translating a part of reality into formulas (if it is reasonable to quantify) and back. You can learn about logical and systematic thinking and abstraction. The full power of course comes first with the tools of calculus. But algebra is a prerequisite for calculus, and for linear algebra.
So we can’t do without algebra for two basic reasons: Without algebra the door to the rest of mathematics is closed, and algebra is a good practice ground for learning the long-term goals of a mathematical education.