The new search field is harder to use than the old one and lacks the possibility to keep the filter active

### Steps to reproduce

1. Where are you starting? What can you see?
Load Element and have lots of rooms and conversions with various people open
2. What do you click?
Click search field
3. More steps…
Oddly, the search field isn't focused, but a modal dialog pops up and one needs to type something there. Confusing UI and stressful to eyes, since one had just stared at the search field and needs to now focus another are in the UI.

### Outcome

#### What did you expect?
The old UI, where search field is a real search field

#### What happened instead?
A (page internal) popup opened

### Operating system

Linux

### Browser information

Firefox 103

### URL for webapp

chat.mozilla.org

### Application version

Element version: 1.11.0

### Homeserver

_No response_

### Will you send logs?

No

Start by reproducing the search interaction in Element with many open rooms and conversations, using the reported Linux and Firefox setup if available. Compare the current behavior with the described old search field; done means clicking the field focuses it directly and allows the filter to remain active without opening a confusing modal.

Written by the indexing model from the issue text.