[Bugfix][Frontend] Apply the model's transcription post-processing on the realtime path - #55786

Purpose

The REST transcription endpoints run model output through post_process_output() /
get_streaming_post_processor_cls(), so models with a structured output format return clean text. The /v1/realtime
WebSocket path forwards output.outputs[0].text verbatim. For Qwen3-ASR every 5 s segment is a separate generation
that starts with a language Chinese<asr_text> header, so realtime clients receive

language Chinese<asr_text>今天天气很好language Chinese<asr_text>我们去公园

instead of the transcript (the "raw format leaks" part of #35767).

Changes:

• SupportsRealtime gains post_process_output() and get_streaming_post_processor_cls() with identity defaults,
  mirroring SupportsTranscription; Qwen3-ASR and Voxtral keep their existing transcription overrides through the MRO
  (Voxtral has none, so its output is unchanged).
• RealtimeConnection._run_generation feeds each delta through a streaming post-processor and starts a fresh one at
  every segment boundary (finish_reason on the completion output). Empty deltas produced while the header is buffered
  are not sent. Token ids fed back to the engine and the usage counters still use the raw tokens.

Related PRs: #49658 also applies the streaming post-processor in connection.py, but with a single instance for the
whole connection; Qwen3ASRStreamingPostProcessor diffs against its accumulated text, so from the second segment on the
language Chinese prefix is emitted again. #51037 strips the header with a private rsplit("<asr_text>") helper as
part of a larger KV-reuse change. This PR is the minimal fix through the existing model hooks and is independent of both.

Test Plan

• New unit test (no GPU) tests/entrypoints/speech_to_text/realtime/test_realtime_post_process.py: pass-through model
  (deltas, token feedback and usage unchanged) and a model using Qwen3ASRStreamingPostProcessor across two segments
  (header stripped from every delta and from done.text, no empty deltas, unstructured output kept).
• End to end: vllm serve Qwen/Qwen3-ASR-1.7B --hf-overrides '{"architectures":["Qwen3ASRRealtimeGeneration"]}',
  FLEURS cmn_hans_cn clips streamed at 100 ms cadence over /v1/realtime, before/after this patch.

Test Result

• Unit test: 3 passed (main@144e79c + this patch).

• E2E on an RTX 5090, main@144e79c ± this patch:

  clip	before	after
  15.5 s	every 5 s segment prefixed with language Chinese<asr_text>, 65 deltas	plain transcript, 46 deltas, no header
  25.4 s (4 segments)	4 prefixed segments, 100 deltas	plain transcript, 71 deltas, no header

  First-delta and done timing relative to audio end are unchanged; usage.completion_tokens is unchanged (65 / 100).
  Segment repetition and mis-transcriptions of the realtime path ([Enhancement]: Qwen3-ASR realtime endpoint produces degraded output — stateless segments, no cross-segment context, raw format leaks #35767 / [Feature][Model] Support stable-window-aware KV reuse for Qwen3-ASR realtime #51037) are not touched by this change.

Developed with AI assistance (Claude); I reviewed every changed line and ran the tests above.