Searching for @user:home.server doesn't show messages of that user

This issue has been migrated from [#14121](https://github.com/matrix-org/synapse/issues/14121).

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### Description

(Forwarded from https://github.com/vector-im/element-meta/issues/1477 )

Searching for @user:home.server doesn't show messages of that user (message sender, author).

### Steps to reproduce

1. Click top-right search button while in a room in element-desktop or element-web. The search bar should appear with the focus on "This room". Enter an account of someone in that room who wrote something previously in the form of `@user:home.server`. Also try `@user` (which IMHO should match everyone with a handle starting like that inside the room), and `user` and `displayname`.
2. Confirm with enter to start the search
3. See that there are no results, or only responses to and mentions of that user (which are fine to include), but not their own messages

### Homeserver

matrix.org

### Synapse Version

1.69.0rc2 (b=matrix-org-hotfixes,3d5242da14)

### Installation Method

_No response_

### Platform

official matrix.org server

### Relevant log output

I'm not running the server so I don't have logs

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

_No response_