Searching channel messages does not find all matching messages (substring match for flag_check inside flag_check.exe ?)

### Description

Re-filed from https://github.com/element-hq/element-web/issues/29365

### Steps to reproduce

1. https://chat.mozilla.org/#/room/#developers:mozilla.org
2. Click info button on the top right
3. Type "flag_check" in the search field

### Outcome

#### What did you expect?

I expected it to find [this message](https://matrix.to/#/!lrZtdjyLpBmoKbMdyx:mozilla.org/$rPzJ40I7clVCt-2StWWXpHcR1mBIuTY2TN5KxUg7ths?via=mozilla.org&via=matrix.org&via=igalia.com).

#### What happened instead?

It doesn't.

### Homeserver

https://mozilla.modular.im/

### Synapse Version

1.124.0

### Installation Method

I don't know

### Database

Unsure

### Workers

I don't know

### Platform

No idea. I don't run the homeserver; I'm simply re-filing the user-experienced issue with the search functionality based on feedback in https://github.com/element-hq/element-web/issues/29365

### Configuration

_No response_

### Relevant log output

```shell
N/A
```

### Anything else that would be useful to know?

The element client POSTs to `_matrix/client/v3/search` with:

```
{"search_categories":{"room_events":{"search_term":"flag_check","filter":{"limit":10,"rooms":["!lrZtdjyLpBmoKbMdyx:mozilla.org"]},"order_by":"recent","event_context":{"before_limit":1,"after_limit":1,"include_profile":true}}}}
```

and the resulting response does not contain the message linked above.

Start by reproducing the POST to _matrix/client/v3/search with search_categories.room_events.search_term set to "flag_check" and the reported room filter. Trace the server-side search handling to determine why the linked message is omitted, then verify that substring matches such as flag_check inside flag_check.exe are returned without breaking existing search behavior.

Written by the indexing model from the issue text.