Synapse 1.150.0 does not expose MatrixRTC transports endpoint despite msc4143_enabled=true

### Description

I am trying to enable MatrixRTC (MSC4143) with Element Call + LiveKit on a self-hosted setup, but the endpoint `/_matrix/client/v1/rtc/transports` is not recognized by Synapse.

### Setup

- Synapse deployed via matrix-docker-ansible-deploy
- Element Web + Element Call
- LiveKit + lk-jwt-service working
- Reverse proxy: Traefik (playbook-managed)

### Configuration

In homeserver.yaml:

```yaml
experimental_features:
msc4143_enabled: true

matrix_rtc:
transports:
- type: livekit
livekit_service_url: "https://rtc.kimiu.fr/livekit/jwt"
```

### Well-known

`https://kimiu.fr/.well-known/matrix/client` returns:
```json
{
"m.homeserver": {
"base_url": "https://matrix.kimiu.fr",
"server_name": "kimiu.fr"
},
"org.matrix.msc4143.rtc_foci": [
{
"type": "livekit",
"livekit_service_url": "https://rtc.kimiu.fr/livekit/jwt"
}
]
}
```

### LiveKit

Health endpoint works:

`curl https://rtc.kimiu.fr/livekit/jwt/healthz` → HTTP 200

### Problem

This request returns 404 / M_UNRECOGNIZED:

```
curl -i https://matrix.kimiu.fr/_matrix/client/v1/rtc/transports \
-H "Authorization: Bearer "
```

Response:

```json
{
"errcode": "M_UNRECOGNIZED",
"error": "Unrecognized request"
}
```

### Expected

Synapse should expose the MatrixRTC transport endpoint when MSC4143 is enabled.

### Notes

- Synapse config clearly includes msc4143_enabled: true
- matrix_rtc block is present in homeserver.yaml
- Synapse has been restarted after config changes

Is there an additional flag or requirement to enable this endpoint?

Thanks

### Steps to reproduce

- list the steps
- that reproduce the bug
- using hyphens as bullet points

### Homeserver

matrix.kimiu.fr (self-hosted)

### Synapse Version

1.150.0

### Installation Method

Docker (matrixdotorg/synapse)

### Database

PostgreSQL (single server, no SQLite migration, no backup restore)

### Workers

Single process

### Platform

Debian 12 (OVH VPS), Synapse running in Docker via matrix-docker-ansible-deploy, Traefik reverse proxy, LiveKit + lk-jwt-service for MatrixRTC

### Configuration

MSC4143 enabled (`experimental_features.msc4143_enabled: true`)

```yaml
matrix_rtc configured with LiveKit:
matrix_rtc:
transports:
- type: livekit
livekit_service_url: "https://rtc.kimiu.fr/livekit/jwt"
```

No custom modules, no unusual presence or message retention settings.

### Relevant log output

No relevant errors found in Synapse logs.

The issue is reproducible via API:

```
curl -i https://matrix.kimiu.fr/_matrix/client/v1/rtc/transports \
-H "Authorization: Bearer "
```

Response:

HTTP/2 404
```
{"errcode":"M_UNRECOGNIZED","error":"Unrecognized request"}
```

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

Element Call standalone (https://call.kimiu.fr) works correctly.

LiveKit and JWT service are reachable and healthy:
`https://rtc.kimiu.fr/livekit/jwt/healthz` → HTTP 200

The `.well-known/matrix/client` endpoint is correctly configured and returns the expected RTC config.

The issue only affects the MatrixRTC integration via Synapse.

Start by checking Synapse's MatrixRTC client API routing and MSC4143 feature-gating for version 1.150.0, using the configured endpoint and homeserver.yaml settings described here. Reproduce the request with curl and trace whether the transport endpoint is registered; done means the endpoint returns the configured LiveKit transport, or the missing requirement is clearly identified.

Written by the indexing model from the issue text.