Paragraphs Browser Widget's doesn't work for base fields (like Paragraphs Library item entity)

Problem/Motivation

Paragraphs browser widgets will cause 404 errors by path /paragraphs_browser/paragraphs_library_item.paragraphs_library_item.paragraphs/name_of_browser/f88be3e4-df64-4dc3-81f3-2c25dab83e4f?_wrapper_format=drupal_ajax.

Steps to reproduce

1. Add browser type by path - /admin/structure/paragraphs_type/browsers/add
2. Add groups for created browser type.
3. Add paragraph types for created groups in the appropriate browser type.
4. Select Paragraphs browser Classic/Paragraphs Browser EXPERIMENTAL widget for paragraphs field on this page - /admin/config/content/paragraphs_library_item/form-display.
5. Select browser type (from step 1) in widget settings and click Save.
6. Add a paragraph on this page - /admin/content/paragraphs/add/default.
7. Check recent log messages and see an error.

Proposed resolution

Write a patch that will fix this issue.

Requirements

Drupal Core: 9.2.3
Paragraphs: 8.x-1.12