Fix README and docs pages that no longer match the code

Bring the READMEs and the docs pages (en and ja) in line with the
current header:

- Fix the client sample that does not compile, the Transfer-Encoding
  statement for default headers, the verifier callback comment, the
  thread pool spawn rule, the sanitize_filename description, and the
  inverted Accept-Encoding sentence.
- Describe recent behaviour: TLS errors behind Read/Write failures,
  Host validation, Connection: close for a body without length,
  content_reader rules, refused CONNECT, WebSocketClient proxies,
  CPPHTTPLIB_INTERIM_RESPONSE_MAX_COUNT, the SSE event size limit, and
  the payload limit of the Stream API.
- Complete the API references of WebSocket, WebSocketClient and the
  stream:: helpers, and the SSE reconnect rules.
- Fix cookbook pages: keychain loading on every backend, received Close
  status, a multipart sample that trusted the client's file name, an
  echo sample that sent binary as text, SNI, pre-request handler scope,
  stop() with WebSocket handlers, and the thread pool notes.
- Restore the mutex in the LLM app's Chapter 5 code.

docs-src/pages/en/cookbook/w01-websocket-echo.md

@@ -21,7 +21,12 @@ int main() {
       if (result == httplib::ws::ReadResult::Fail) {
         break;
       }
-      ws.send(msg); // echo back what we received
+      // echo back what we received, in the same frame type
+      if (result == httplib::ws::ReadResult::Binary) {
+        ws.send(msg.data(), msg.size());
+      } else {
+        ws.send(msg);
+      }
     }
   });
 
@@ -29,7 +34,7 @@ int main() {
 }
 ```
 
-Register a WebSocket handler with `svr.WebSocket()`. By the time the handler runs, the WebSocket handshake is already complete. Inside the loop, just `ws.read()` and `ws.send()` to get a working echo.
+Register a WebSocket handler with `svr.WebSocket()`. By the time the handler runs, the WebSocket handshake is already complete. Inside the loop, just `ws.read()` and `ws.send()` to get a working echo. A binary message goes back through the `data()` + `size()` overload so it stays binary (more on that below).
 
 The `read()` return value is a `ReadResult` enum:
 
@@ -76,7 +81,7 @@ The `std::string` overload sends as **text**; the `const char*` + size overload
 
 ## Thread pool implications
 
-A WebSocket handler holds its worker thread for the entire life of the connection — one connection per thread. For many concurrent clients, configure a dynamic thread pool.
+A WebSocket handler holds its worker thread for the entire life of the connection — one connection per thread. The default pool already grows under load, up to 4× its base thread count. For many concurrent clients, raise that maximum.
 
 ```cpp
 svr.new_task_queue = [] {