The Play Store will not work better unless its sister app, Google Play Services, is also updated to its final KitKat build.
Now, try to download a small, lightweight app like "Google Opinion Rewards" or a simple file manager to confirm the Play Store is functioning correctly.
You need a version of the Play Store that supports API 19 (Android 4.4+). Based on records, versions around (from 2022) are some of the last to explicitly support KitKat.
Final note: If your device has Android 4.4.2 and you need a "better" experience, consider flashing a custom ROM like LineageOS 14.1 (Android 7.1) if available for your device. That is the true long-term solution.
This guide will walk you through the entire process, from finding the correct APK files to optimizing your older device for better performance.