Phone Link on Windows can't see Gmail contacts // Outlook Mail on Android fails to display Gmail

1. On my Windows 10 Pro laptop:
   1. As of a couple of weeks (don't know exactly when), the Phone Link app stopped identifying the names of message senders which are listed in the connected Gmail account, but shows those listed in my Hotmail account.
   2. When I wish to send a new message from that app and search for a contact by name, it doesn't find it if it is only listed in my Gmail account.
   3. The Contacts app (on Windows), v. 10.2202.31.0, shows all my contacts.
2. On my smartphone:
   1. All contacts (Hotmail, Gmail) show in the Contacts app and in the Messaging one.
   2. Additionally, Outlook Mail stopped working correctly with my Gmail account at about the same time: it doesn't display a connection problem, but doesn't show any incoming email as of the beginning of the month. It shows many mail from 5/31 back, even though I have deleted them from my Gmail account (by going directly to gmail.com on a browser). The Gmail security params show that Outlook is allowed to access messages. Sending messages (with Outlook Mail, through this Gmail account) works fine. I have uninstalled/reinstalled it a couple of times: to no avail. The Gmail app works fine both ways.

I'd be very glad for any help with this puzzling problem.