Core GMS package Suddenly started Malfunctioning (old Huawei Honor 8, ~2 years in use)
The issue was resolved. Since only certain Google-related apps were affected, I was able to back up everything else I had on the phone, including data and apps. After that I ran the Factory Data Reset and used my back up to make the process of setting up the phone and all the apps, I had been using, easier. The problem was resolved immediately after the reboot and I was able to set up the phone with my Google account.

/Posting this answer to end the thread/

P.S.: Prior to running a factory reset, I attempted clearing cache and data for the affected and related apps and resetting phone settings without deleting all data. That did not work.