Why there is no any dependent dropdown widget for taxonomy hiararchical select in views exposed block?

It means that it would be nice if you could follow the documentation on the Priority and Category fields in the future.
Please also take the time to write a clear description of what you want added or changed. I don't understand at all what you are trying to say.

Ah, so you want a taxonomy term filter for search views where you first select the term from the highest level in the vocabulary, and can then drill down to more specific terms? I haven't seen that in use in a normal view, but I can imagine it and it does sound practical for some use cases.
In my opinion, however, it would be too specific for the Search API, which should stay as generic as possible. It could be included in another module, though.

As a workaround, you can just use facets. You have to activate the "Index hierarchy" data alteration for the field and then create a facet for it for which you set "Flatten hierarchy" to "No". You could then also re-style the facet block to look like you vision your exposed filter to look.

Yes, they can. As said: You have to activate the "Index hierarchy" data alteration for the field and then create a facet for it for which you set "Flatten hierarchy" to "No". You could then also re-style the facet block to look like you vision your exposed filter to look.
Which part of this isn't working?

And you have an existing search on the front page? Otherwise, neither exposed filters nor facets would be readily possible, but you could use the "Facets block" Views display.

I'm sorry, but I don't understand your last comments at all. Please try to rephrase, or maybe let someone with better English translate for you.

Ah, OK, now I understand! So you need some kind of Javascript mechanism to display the second level of options, without a new page load …
Then sorry, I don't think this is possible without some custom code, neither in Views nor with facets. Solving it in Views might be easier then, but I really can't say. But, in any case, you'd need some custom code to be written.

I didn't know of that module, no, but basically an integration should be possible. Maybe that would be the easiest (or cleanest) way to solve your problem. However, coding would still be required, if you don't find someone who wants to implement this for you.

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I've the same issue here, but will try a workaround using views and custom Js.

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This issue has not seen activity in over 2,5 years. I am therefore closing this issue to clean up the issue queue. Feel free to re-open and update this issue if you feel this issue is still relevant and of importance.