On the contrary, it's irrelevant if you're not writing a sorting algorithm or your main role doesn't consist of writing any algorithms. Most web developers I know simply call sort() in whatever language they're using instead of reinventing the wheel. Even my work in Objective-C where I handle retaining and releasing memory, I still call sort() (and compare()) without caring about algorithm efficiency.
If you're calling a sorting algorithm, you should still know about the efficiency of that sorting algorithm takes so you can make a choice between sorting an array or indexing using a data structure (which data structure).
Web is just a UI to layer software that _does something_, algorithms are a way to get that something done. Have you ever needed to understand why e.g., a SQL JOIN is taking too long? That's an example where algorithms matter (in selecting the index type, knowing whether to use a join between two tables or to denormalize etc...)
Knowing how to do your own memory management and knowing Objective C is great and valuable, but if mobile/Mac desktop application development were to become less valuable and more commodity (right now, I don't think it will happen-- but I can't predict the market), will you be able to apply these skills to (say) working on more complex C and C++ software such as a web browser?
IMO, I think you're spending more time talking about edge cases than the norm. How many people are web developers vs those developing web browsers?
Understanding query cost has more to do with data structures than algorithms. Why does the optimizer seek vs scan? What column(s) do I need to index and how should it be structured? You really don't need to understand how things get sorted in a b-tree.
Knowing normal forms is also different from algorithms. Ergo, it would be more useful to ask "When would you denormalized your data?" instead of "What's the difference between Quicksort and Bubble Sort?"
Objective-C is really just one of many languages I know and use. It just happens to be the most recent. Oddly enough I did spend some time (many years ago) implementing my own browser for a job. We used Java and, again, I never really had to know the complexities of sorting algorithms.
