[PickerInput]: can't select items if datasource api return list only if search is typed

## Description (optional)

If datasource api configured in a such way when api return items only when search is typed, if not - api return empty array. In this case item check became broken, check is not react on click.

It seams that issue related to the DataSource tree load behavior - if search is typed we load items only in visible tree, but checking work based on full tree, which is empty because when search is empty we return empty list.

## Steps to Reproduce

1. Open sandbox - https://codesandbox.io/p/sandbox/uui-forked-z4cm8w?file=%2FExample.tsx%3A10%2C42&workspaceId=1dc5a6c7-6fb2-4cae-8ed5-c1bd7e6e64f0
2. Type to search more than 3 symbols
3. Try to check item

## Actual result

Item isn't checked

## Expected result

Item became checked

## Environment

* UUI version: 5.11.0