What books are there that cover elementary algebra, but are not bloated with pedagogically padding?.

Take a look at Serge Lang's Basic Mathematics. It may be slightly more basic than you would probably like but it is nonetheless a great book. It is based on deriving and proving things from previously introduced concepts instead of just introducing things seemingly out of the blue and then spending several pages motivating the uses and talking endlessly about history. Surely there is some pedagogical method to it but it is aimed at people who find the topic interesting already and can appreciate the mathematics without seeing pictures of the space ship launch or elaborate 3d scenes involving cubes and spheres.
I would put it at a level somewhere between high school and college. I'm studying it right now but since I have very little true knowledge and understanding of mathematics from high school I find it a bit challenging in places, e.g. we never proved a thing in school so I had/have to learn how to go about proving things. I have learned so much about mathematics and proofs and how things really work, and I am only at the start of the book so there is quite a journey ahead of me!
Other books that may be of interest to you: Gelfand's Algebra, Method of coordinates, Functions and graphs, and Trigonometry. I've read some great things about them and I work through Algebra alongside Lang's text and I find them to complement each other fairly well. However, there is no coverage of complex numbers in this one. They are all very short and packed with great insight and challenging problems.

I'm not 100% sure that this would fit your needs, but the Art of Problem Solving books series by Richard Rusczyk and Mathew Crawford go very indepth and have little "padding".