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Because it's all abstracted away. B-trees are used in the underlying implementations of RDBMS. Other data stores such as MongoDB, Memcached, and Redis use hash tables.

Arrays, linked lists, and dictionaries are usually abstracted away by whatever scripting language is in use.

I can't think of common use cases for data structures like heaps/priority queues and graphs. Both come in handy for dependency management and loading assets from disk, like in the case of writing a package manager or a game. But nothing that complex is needed for your typical web app.

Code in web applications feels shoddily written, without a kind of bigger picture. For example if you display a product in the interface, why don't you have a data structure representing a product? Or if you have a form for creating a new product, why don't you have a data structure representing that?

The code feels shoddily written because it is shoddily written. Well written web applications do use these "data structures"; its just that they're called "models" rather than data structures.