ci: Check binaries separately from building them (#7355)

Co-authored-by: Copilot Autofix powered by AI <175728472+Copilot@users.noreply.github.com>

docker/test_files/cpp_sources/tsan.cpp

@@ -0,0 +1,26 @@
+#include <iostream>
+#include <thread>
+
+static int kCounter = 0;
+
+void
+increment()
+{
+    for (int i = 0; i < 100'000; ++i)
+    {
+        ++kCounter;
+    }
+}
+
+int
+main()
+{
+    std::thread t1(increment);
+    std::thread t2(increment);
+
+    t1.join();
+    t2.join();
+
+    std::cout << "Final counter value: " << kCounter << std::endl;
+    return 0;
+}