[core] Inline api_is_connected() for hot-path callers

Move the api_is_connected() definition from util.cpp to util.h and mark it
ESPHOME_ALWAYS_INLINE. The body is trivial — a nullptr check on
global_api_server plus APIServer::is_connected() (which is
!clients_.empty()) — so the out-of-line call8 was pure overhead for hot
paths that check connectivity every loop tick (e.g. zwave_proxy::loop,
serial_proxy::loop).

With USE_API disabled the function collapses to "return false" at compile
time and folds away entirely.

util.h now pulls in api_server.h under USE_API. Only 25 .cpp files include
util.h and all of them are on USE_API builds in practice (wifi, safe_mode,
nextion, web_server, etc.), so the additional transitive include is
essentially free on the platforms that matter.

Measured on ESP32-S3: the call8 in ZWaveProxy::loop() is replaced by
5 inline Xtensa instructions (load global_api_server, deref, null check,
clients_ start/finish compare) — no call overhead, no stack frame.

esphome/core/util.cpp

@@ -4,25 +4,12 @@
 #include "esphome/core/version.h"
 #include "esphome/core/log.h"
 
-#ifdef USE_API
-#include "esphome/components/api/api_server.h"
-#endif
-
 #ifdef USE_MQTT
 #include "esphome/components/mqtt/mqtt_client.h"
 #endif
 
 namespace esphome {
 
-bool api_is_connected() {
-#ifdef USE_API
-  if (api::global_api_server != nullptr) {
-    return api::global_api_server->is_connected();
-  }
-#endif
-  return false;
-}
-
 bool mqtt_is_connected() {
 #ifdef USE_MQTT
   if (mqtt::global_mqtt_client != nullptr) {

esphome/core/util.h

@@ -1,10 +1,28 @@
 #pragma once
 
 #include <string>
+
+#include "esphome/core/defines.h"
+#include "esphome/core/helpers.h"
+
+#ifdef USE_API
+#include "esphome/components/api/api_server.h"
+#endif
+
 namespace esphome {
 
-/// Return whether the node has at least one client connected to the native API
-bool api_is_connected();
+/// Return whether the node has at least one client connected to the native API.
+///
+/// Inline so that hot-path callers (e.g. component loop() ticks that check connectivity every
+/// iteration) can skip the call8/return pair. With USE_API disabled this trivially returns false
+/// and collapses at compile time.
+#ifdef USE_API
+ESPHOME_ALWAYS_INLINE inline bool api_is_connected() {
+  return api::global_api_server != nullptr && api::global_api_server->is_connected();
+}
+#else
+ESPHOME_ALWAYS_INLINE inline bool api_is_connected() { return false; }
+#endif
 
 /// Return whether the node has an active connection to an MQTT broker
 bool mqtt_is_connected();