Hi @David Reinhardt ,
This can be achieved. Per my test,in SharePoint Admin center there is currently not supported to set people with existing access by default, however we can set it through powershell.
Run this script(change to your information) using Windows Powershell ISE:
$Username='xxx@xxxx.onmicrosoft.com'
$Password = 'xxxxx'
#region Credentials
[SecureString]$SecurePass = ConvertTo-SecureString $Password -AsPlainText -Force
[System.Management.Automation.PSCredential]$PSCredentials = New-Object System.Management.Automation.PSCredential($Username, $SecurePass)
#endregion Credentials
Connect-SPOService -Url https://company-admin.sharepoint.com -credential $PSCredentials
Set-SPOSite -identity https://company.sharepoint.com/sites/yoursite -DefaultLinkToExistingAccess 1
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