test: Use Synapse with Simplified Sliding Sync in our integration tests

[Hywan]

In #3676, we are migrating the SDK to use Simplified Sliding Sync instead of Sliding Sync itself.

When the PR was written, the version of Synapse used in our integration tests didn't have Simplified Sliding Sync, but was still using the Sliding Sync Proxy. As rightly suggested by @bnjbvr in #3676 (comment), we should not forget to move away from the Proxy once everything is ready inside Synapse.

Step 1

Run tests with Synapse instead of the sliding sync proxy:

• test: Bye bye SS proxy, hello Synapse \o/ #3983

Step 2

We now need to migrate the following tests (that are failing since the migration to Synapse):

• sliding_sync::room::test_room_avatar_group_conversation
• sliding_sync::room::test_room_notification_count
• sliding_sync::room::test_delayed_decryption_latest_event
• sliding_sync::room::test_room_info_notable_update_deduplication
• sliding_sync::room::test_room_preview
• sliding_sync::notification_client::test_notification
  • Blocked by Sliding sync returns initial: false for a first response element-hq/synapse#17754
  • The fix test(integration): Enable test_notification #4029

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• Address [meta] SSS & PPS #3647