Users who register through GitHub cannot bind their email

This issue has been migrated from [#14326](https://github.com/matrix-org/synapse/issues/14326).

---

### Description

When a user signs up through GitHub, the user cannot bind their email address and cannot back up the encryption keys.
It shows
But if you are a registered user via email, it's perfectly normal to do all this.

### Steps to reproduce

- Register via Github
- Bind a email address or backup the encryption keys

### Homeserver

hashi.sbs

### Synapse Version

Synapse 1.70.1

### Installation Method

Debian packages from packages.matrix.org

### Platform

Ubuntu 20.04.5
Python 3.8

### Relevant log output

```shell
No log
```

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

Related config file snippets
```
enable_registration_without_verification: false
registrations_require_3pid:
- email

oidc_providers:
- idp_id: github
idp_name: Github
idp_brand: "github" # optional: styling hint for clients
discover: false
issuer: "https://github.com/"
client_id: "***" # TO BE FILLED
client_secret: "***" # TO BE FILLED
authorization_endpoint: "https://github.com/login/oauth/authorize"
token_endpoint: "https://github.com/login/oauth/access_token"
userinfo_endpoint: "https://api.github.com/user"
scopes: ["read:user"]
user_mapping_provider:
config:
subject_claim: "id"
localpart_template: "{{ user.login }}"
display_name_template: "{{ user.name }}"

enable_3pid_changes: true
```

Re-login did not solve the problem.

Start by reproducing the issue with the listed GitHub OIDC configuration on Synapse 1.70.1, then trace the email-binding and encryption-key backup flows for users registered through GitHub. Confirm whether the same flows work for email-registered users and identify the missing behavior before defining a fix.

Written by the indexing model from the issue text.