SearchHandler - Flickering issue and keyboard not dismissed (still happening in 9.0.90)

### 🔁 Context

This is a follow-up to: [https://github.com/dotnet/maui/issues/30072](https://github.com/dotnet/maui/issues/30072)
Although the original issue was closed after PR #30222, the problem is still present in version **9.0.90**.

### 📱 Device

* Xiaomi Redmi Note 9 Pro
* Android 10 - API 29

### ❗Current issues

1. **Flickering when typing in `SearchHandler`**

* The keyboard opens and closes repeatedly while typing.
* Characters are lost as a result.
* Clearly shown in the attached video.

2. **Keyboard remains visible after losing focus**

* When navigating to another page (for example, selecting a result), the `SearchHandler` loses focus but **the keyboard stays open**, which shouldn't happen.

### 🎥 Video

See attached video with both issues:

* Part 1: flickering during typing.
* Part 2: keyboard remains after losing focus and navigating.

https://github.com/user-attachments/assets/f57388b0-e76b-464b-b145-495cdfccda6d

Start by reproducing the SearchHandler behavior on the listed Xiaomi Redmi Note 9 Pro running Android 10/API 29 with version 9.0.90, using the attached video as the expected scenario. Trace typing and navigation after focus loss; done means typing no longer flickers or drops characters and the keyboard dismisses when SearchHandler loses focus.

Written by the indexing model from the issue text.