It is your choice which career to pursue, but in my experience, absolute majority of programmers don't know algorithms and data structures outside of very shallow understanding required to pass some popular interview questions. May be you've put too high artificial barriers, which weren't necessary.
To be a professional software developer, you need to write code to solve real life tasks. These tasks mostly super-primitive in terms of algorithms. You just glue together libraries and write so-called "business-logic" in terms of incomprehensible tree of if-s which nobody truly understands. People love it and pay money for it.
Should I be familiar with every step of Dijkstra’s search algorithm and remember the pseudocode at all times? Why don’t the textbooks explain why the algorithm is correct?
Somehow, I think you already know the answer to that is "no".
I've been working as a software engineer for over 8 years, with no computer science education. I don't know what Dijkstra's search algorithm is, let alone have memorised the pseudocode. I flicked through a book of data structures and algorithms once, but that was after I got my first software job. Unless you're only aiming for Google etc, you don't really need any of this.