I am having trouble adding contacts to my favorite list. Some of them are even disappearing and I get a message telling me to sign in to Google.

What is your default Contacts account? If none, please add an email account and enable contacts.

See the pic below

Unlike some Android devices, contacts are neither stored locally on an iPhone nor in SIM. They are synced to an email account that you had Google, iCloud, AOL, Exchange, Outlook, Yahoo or any Other

If you have a corporate email account added through a profile on your device, please check with your organization's system administrator.

After you tap Add to Favourites after opening the Contact, three options will pop up.

Select the Phone section and select the Phone number. Watch the GIF below.