Hi @Andy,
SharePoint 2016 doesn’t offer the handy “Sign-in as Different User” feature under the welcome menu.
Apart from the "clear the cookie of our browser or use a private mode" you said. Here are two more ways for your reference:
1, Navigate to Close Connection Page
Navigate to the URL: https://Your-Site/_layouts/closeConnection.aspx?loginasanotheruser=true
In my environment, I use this url.
http://sp16/sites/amy/_layouts/closeConnection.aspx?loginasanotheruser=true
2, Edit “Welcome.ascx” file (modifying files in the SharePoint file system is not a good way)
Log in to Windows with the account configured for SharePoint Server to edit “Welcome.ascx” file, which is at “C:\Program Files\Common Files\microsoft shared\Web Server Extensions\16\TEMPLATE\CONTROLTEMPLATES” folder.
Open it in notepad, Add this new entry under the node: SharePoint:FeatureMenuTemplate
<SharePoint:MenuItemTemplate runat="server" ID="ID_LoginAsDifferentUser"
Text="<%$Resources:wss,personalactions_loginasdifferentuser%>"
Description="<%$Resources:wss,personalactions_loginasdifferentuserdescription%>"
MenuGroupId="100"
Sequence="100"
UseShortId="true"
/>
Like this:
Save the file (No need for IIS Reset!) That’s it! Refresh the page, you can see the “Sign in as different user” menu item now!
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.