Hi. I am following up with a fix to this issue. I tried the steps below and was successfully able to sync my contacts. The instructions that I have written up below explain the process:
The Android 10 watches do not come with the Google Contact Sync service automatically installed. It must be installed manually. As an extra measure, please also download the Google Contacts app from the Google Play Store. For the complete steps, see below:
(1) Download the Google Contact Sync app for Android 10 from APKMirror or for easier use, you can also use my Google Drive link https://drive.google.com/file/d/1rwo7qpudBcBEwd7lq29aWHDQ-u_JzFPH/view?usp=drivesdk (your choice).
(2) Install the app and then go to Settings\Apps and Notifications.
(3) Tap on show all apps installed on your device.
(5) Tap on Google Contact Sync, and give it permissions to read your contacts in the permissions area.
(6) Now download Google Contacts from the Play Store, https://play.google.com/store/apps/details?id=com.google.android.contacts&hl=en_US&gl=US
(7) After Google Contacts gets installed, tap on your Google account. This should now allow your contacts to sync automatically from your Google account.
I hope this is useful to other people.