Are there any current, major phones based on Android 4.4 that don't have Google Chrome pre-installed as a browser? I'm asking since Android 4.4 doesn't include a browser and it needs to be licensed separately by the handset manufacturer.
Highly doubtful. If the phone is using Google Play Services, which every manufacturer must use and agree to a strict set of terms and conditions if they want their device to access the Google Play Store, then Chrome will be installed. 

It's one of Google's key apps, along with Gmail, Maps and so on.