Replace native input with Input component in search component (docs)

[antonina220590]

📝 Describe the refactoring task

In the search component replace the standard HTML <input> tag with our shared reusable <SearchInput /> component

📍 Code location

Search component - src/app/docs/components/search
SearchInput reusable component - src/shared/ui/search-input

✅ Definition of Done (DoD):

• The <input> tag is no longer used in the search.tsx file.
• The <Input /> component is used instead, with all necessary props (value, onChange, placeholder, etc.).
• All functionality of the old input is preserved.