Dear Carrie,
Whether you select Send to>Email or Promote>Email, in both the cases it will send an email to the recipient. If you type a Microsoft 365 group name in Send To: field, the members will receive the email in the group inbox and if you type an individual’s email address, the recipient will receive the email in their personal inbox.
As you mentioned, you have tried both the above options and neither worked may I know if you see any email in your Sent items of your Outlook?
Besides, may I know if you have tried sharing the page/news from Page library? For this, go to Site Contents, click Site Pages, select the page that you wish to share, click Share or Copy link on the ribbon to share the page. Please note, “Anyone” option will be greyed out as pages are not allowed to be shared with anonymous permission.
Important: Recipients of the page can view only the content or web part which they have access to. For example, if you have added a document library webpart to the site page, the said web part will not appear to the Recipient if they don’t have access to the library explicitly, instead they will see a blank space at that place.
If you need any help, let me know.
Regards,
Neha