Newer versions of Play Services are designed for Android 12+ and will not install on KitKat.

Since the Play Store often fails on legacy devices, manual installation is the most reliable method. Google Play services (Android 4.4+) APKs - APKMirror

You must sideload the "final" compatible version of the APK specifically built for API 19 (Android 4.4+). Finding the Right Version for Your Hardware

Most smartphones and tablets from that era. x86: Intel-based tablets.