[api] Fix review comments and add USE_API guards

- Update comments to reflect that ProtoService methods moved to
  APIConnection, not APIServerConnectionBase
- Fix comment referring to read_message as "override"
- Wrap #include "api_connection.h" and read_message_ implementation
  in #ifdef USE_API guards

script/api_protobuf/api_protobuf.py

@@ -2947,7 +2947,9 @@ namespace esphome::api {
     cpp = FILE_HEADER
     cpp += """\
 #include "api_pb2_service.h"
+#ifdef USE_API
 #include "api_connection.h"
+#endif
 #include "esphome/core/log.h"
 
 namespace esphome::api {
@@ -3051,11 +3053,12 @@ static const char *const TAG = "api.service";
                 result += "#endif\n"
         return result
 
-    # Generate read_message as APIConnection method (not base class) so the compiler
+    # Generate read_message_ as APIConnection method (not base class) so the compiler
     # can devirtualize and inline the on_* handler calls within the same class.
-    # APIConnection declares the override in api_connection.h.
+    # APIConnection declares this method in api_connection.h.
 
-    out = "void APIConnection::read_message_(uint32_t msg_size, uint32_t msg_type, const uint8_t *msg_data) {\n"
+    out = "#ifdef USE_API\n"
+    out += "void APIConnection::read_message_(uint32_t msg_size, uint32_t msg_type, const uint8_t *msg_data) {\n"
 
     # Auth check block before dispatch switch
     out += "  // Check authentication/connection requirements\n"
@@ -3100,6 +3103,7 @@ static const char *const TAG = "api.service";
     out += "      break;\n"
     out += "  }\n"
     out += "}\n"
+    out += "#endif  // USE_API\n"
     cpp += out
     hpp += "};\n"