Realtime models: OTel span misses input/output or usage on `realtime_inference` span, no user input on `agent_turn` span

Bug Description

With a realtime model (tested with GPTLiveModel, same code path as RealtimeModel), the spans from livekit-agents 1.8.x are missing most of the conversation content. A GenAI backend shows every turn as an empty generation with no cost.

The cascaded pipeline path is fine. llm_request gets gen_ai.input.messages, gen_ai.output.messages and usage, and agent_turn gets lk.pii.instructions and lk.pii.user_input. The realtime path never calls the content helpers.

What's missing

The user's transcript is on no span at all. Only user_speaking debug spans, without text.

Usage: GPT-Live's backend model (gpt-5.6-luna, ~1.4k prompt / 74 completion tokens per turn) is emitted as LLMMetrics on metrics_collected but never attached to a span. The voice model's RealtimeModelMetrics reports zero tokens (billed by duration), so realtime_inference has no usage even when record_realtime_metrics matches.

Expected Behavior

Parity with the pipeline path: realtime_inference carries input/output messages, system instructions and usage; agent_turn carries lk.pii.user_input and lk.pii.instructions.

Happy to send a PR. Rough shape: reuse to_input_messages / to_output_messages from telemetry/gen_ai.py on the realtime chat context, and attach the backend LLMMetrics to the pending realtime_inference span by response.id.

Reproduction Steps

1. `AgentSession(llm=GPTLiveModel(...))` with one function tool, `set_tracer_provider(provider)` with any OTLP exporter. `OTEL_INSTRUMENTATION_GENAI_CAPTURE_MESSAGE_CONTENT` unset (defaults on), no redaction.
2. Short conversation with one tool call.
3. Look at the spans.

Real example, the Langfuse voice demo (https://langfuse.com/docs/demo) runs exactly this setup. Public trace: https://cloud.langfuse.com/project/clkpwwm0m000gmm094odg11gi/traces/738d3c599f6071781340ebe0ead4054c?observation=ca5d2ea847dfef01

`realtime_inference` there carries only `gen_ai.operation.name=generate_content`, `gen_ai.provider.name=openai`, `gen_ai.request.model=gpt-live-1`, `gen_ai.request.stream=true`, `gen_ai.output.type=speech`, `gen_ai.conversation.id`. `agent_turn` has `lk.pii.response.text` and nothing about the user.

Operating System

MacOS

Models Used

No response

Package Versions

livekit-agents 1.8.1, livekit-plugins-openai 1.8.1, exporter is the Langfuse Python SDK. Same on `main` today.

Session/Room/Call IDs

No response

Proposed Solution

Additional Context

No response

Screenshots and Recordings