Incorrect HTML responses when providing possibly non existent matrix identifiers

This issue has been migrated from [#15392](https://github.com/matrix-org/synapse/issues/15392).

---

### Description

Calling APIs with (possibly) non existent matrix identifiers (like: `@alfotest:some`) triggers non json responses from the home server with HTTP errors 502.

Examples of such APIs are:
- [Get profile](https://spec.matrix.org/v1.6/client-server-api/#get_matrixclientv3profileuserid)
- [Create Room](https://spec.matrix.org/v1.6/client-server-api/#post_matrixclientv3createroom)(with at least one invited user)
- [Invite](https://spec.matrix.org/v1.6/client-server-api/#post_matrixclientv3roomsroomidinvite)

⚠️ The issue is particularly serious with the create room API (eg. with "invite": [`@alfotest:some`]) since even if the clients gets back an HTTP 502 and a HTML response, the room creation actually succeeds.

### Steps to reproduce

- Open https://matrix-client.matrix.org/_matrix/client/v3/profile/@alfotest21:some

### Homeserver

matrix.org

### Synapse Version

{"server_version":"1.80.0 (b=matrix-org-hotfixes,ab0a5f1972,dirty)","python_version":"3.8.12"}

### Installation Method

I don't know

### Database

I don't know

### Workers

I don't know

### Platform

I don't know

### Configuration

I don't know

### Relevant log output

```shell
None
```

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

_No response_

Start by reproducing the profile request for @alfotest21:some, then compare the Get profile, Create Room with an invited user, and Invite API entry points. Trace how a possibly nonexistent Matrix identifier produces the HTTP 502 and HTML response; done means these APIs return the appropriate Matrix API response and do not report room creation as failed when it succeeds.

Written by the indexing model from the issue text.