I have seen about 5 other very recent posts from people who are having the exact same problem and not one person has received a satisfactory response. Here goes again. I don't think this is a "run-the-troubleshooter" issue. There is almost certainly a bug or a very obscure configuration setting that needs better visibility. I am very technical - worked at Microsoft in the product groups for 10 years and am a developer. Someone from Microsoft should investigate this.
When I attempt to send an email using Windows 10 Mail on the PC or Outlook Mail on Windows 10 Phone and I type the name of someone I want to send email to, the app can not seem to find them - even though they are clearly visible in the People app. I've tried typing the display name (friendly name) and the person's email address. Neither work. If I choose the "Search Directory" option, it won't find the people either. I have checked the security settings for Contacts and permissions are granted to Mail and Calendar. I am able to send email to some contacts but I can't figure out why some get found and others don't. Outlook 2016 works perfectly. This worked just fine on Windows Phone 8.1 and seems to be broken now.
Some specifics of my configuration: I use Exchange 365 for email/calendar/contacts and sign in using a "Business" Microsoft Account of the form *** Email address is removed for privacy ***. I have a regular Microsoft Account with the exact same name *** Email address is removed for privacy ***. This can be kind of confusing - one account is labeled the Outlook account and the other is labeled the MyDomain account. Things are configured in Mail 10 to sync the MyDomain/Exchange email/calendar/contacts. The Outlook account is configured to not sync Email. I don't use the Outlook account for email/calendar/contacts at all and this is fine. I wish I could completely hide this account from the calendar and mail but you can't get rid of it presumably because it is the Microsoft Account account used for OneDrive and logging on to Windows.