Print full startup/initialization error (#15569)

I found the error in the **Before** really vague and obtuse and didn't realize port `5432` corresponded to the Postgres port until searching the codebase. It says to check the logs but that wasn't my first instinct. It's just more obvious if we just print the full thing which gives context of the error type and the traceback to the relevant area of code.

#### Before

```
$ poetry run python -m synapse.app.homeserver -c homeserver.yaml
**********************************************************************************
 Error during initialisation:
    connection to server at "localhost" (::1), port 5432 failed: Connection refused
 	Is the server running on that host and accepting TCP/IP connections?
 connection to server at "localhost" (127.0.0.1), port 5432 failed: Connection refused
 	Is the server running on that host and accepting TCP/IP connections?
 
 There may be more information in the logs.
**********************************************************************************
```

#### After

```sh
$ poetry run python -m synapse.app.homeserver -c homeserver.yaml
**********************************************************************************
 Error during initialisation:
     Traceback (most recent call last):
       File "/home/eric/Documents/github/element/synapse/synapse/app/homeserver.py", line 352, in setup
         hs.setup()
       File "/home/eric/Documents/github/element/synapse/synapse/server.py", line 337, in setup
         self.datastores = Databases(self.DATASTORE_CLASS, self)
       File "/home/eric/Documents/github/element/synapse/synapse/storage/databases/__init__.py", line 65, in __init__
         with make_conn(database_config, engine, "startup") as db_conn:
       File "/home/eric/Documents/github/element/synapse/synapse/storage/database.py", line 161, in make_conn
         native_db_conn = engine.module.connect(**db_params)
       File "/home/eric/.cache/pypoetry/virtualenvs/matrix-synapse-xCtC9ulO-py3.10/lib/python3.10/site-packages/psycopg2/__init__.py", line 122, in connect
         conn = _connect(dsn, connection_factory=connection_factory, **kwasync)
     psycopg2.OperationalError: connection to server at "localhost" (::1), port 5432 failed: Connection refused
     	Is the server running on that host and accepting TCP/IP connections?
     connection to server at "localhost" (127.0.0.1), port 5432 failed: Connection refused
     	Is the server running on that host and accepting TCP/IP connections?
 
 
 There may be more information in the logs.
**********************************************************************************
```

2 files changed, 7 insertions, 1 deletions

diff --git a/changelog.d/15569.feature b/changelog.d/15569.feature
new file mode 100644
index 0000000000..b58af8ad55
--- /dev/null
+++ b/changelog.d/15569.feature
@@ -0,0 +1 @@
+Print full error and stack-trace of any exception that occurs during startup/initialization.
diff --git a/synapse/app/_base.py b/synapse/app/_base.py
index 7f83b34d89..4dfcf484fa 100644
--- a/synapse/app/_base.py
+++ b/synapse/app/_base.py
@@ -21,6 +21,7 @@ import socket
 import sys
 import traceback
 import warnings
+from textwrap import indent
 from typing import (
     TYPE_CHECKING,
     Any,
@@ -212,8 +213,12 @@ def handle_startup_exception(e: Exception) -> NoReturn:
     # Exceptions that occur between setting up the logging and forking or starting
     # the reactor are written to the logs, followed by a summary to stderr.
     logger.exception("Exception during startup")
+
+    error_string = "".join(traceback.format_exception(e))
+    indented_error_string = indent(error_string, "    ")
+
     quit_with_error(
-        f"Error during initialisation:\n   {e}\nThere may be more information in the logs."
+        f"Error during initialisation:\n{indented_error_string}\nThere may be more information in the logs."
     )