Include examples in type checking

Remove `examples/` from the `pyrightconfig.json` ignore list and fix
the resulting type errors across all example files. Common fixes:

- Required API keys: `os.getenv("X")` -> `os.environ["X"]` so the
  return type is `str` rather than `str | None`, and misconfiguration
  fails fast.
- Narrow `LLMContextMessage` union members with `isinstance(..., dict)`
  before dict-style access.
- `assert isinstance(params.llm, ...)` before calling service-specific
  methods that aren't on the base `LLMService`.
- Guard optional frame fields (e.g. `LLMSearchResponseFrame.search_result`)
  before use.

examples/video-processing/video-processing.py

@@ -96,20 +96,13 @@ Respond to what the user said in a creative and helpful way. Keep your responses
 
 async def run_bot(pipecat_transport):
     llm = GeminiLiveLLMService(
-        api_key=os.getenv("GOOGLE_API_KEY"),
+        api_key=os.environ["GOOGLE_API_KEY"],
         voice_id="Puck",  # Aoede, Charon, Fenrir, Kore, Puck
         transcribe_user_audio=True,
         system_instruction=SYSTEM_INSTRUCTION,
     )
 
-    messages = [
-        {
-            "role": "developer",
-            "content": "Start by greeting the user warmly and introducing yourself.",
-        }
-    ]
-
-    context = LLMContext(messages)
+    context = LLMContext()
     user_aggregator, assistant_aggregator = LLMContextAggregatorPair(
         context,
         user_params=LLMUserAggregatorParams(vad_analyzer=SileroVADAnalyzer()),
@@ -141,6 +134,12 @@ async def run_bot(pipecat_transport):
     async def on_client_ready(rtvi):
         logger.info("Pipecat client ready.")
         # Kick off the conversation.
+        context.add_message(
+            {
+                "role": "developer",
+                "content": "Start by greeting the user warmly and introducing yourself.",
+            }
+        )
         await task.queue_frames([LLMRunFrame()])
 
     @pipecat_transport.event_handler("on_client_connected")