macOS – Invitations sent from Calendar not being received

I want to send invites from an event in Calendar to another person. Their contact details (name, email and mobile) are in Contacts and are correct. My own account details are all correct and enabled.

I start to type the person's name into the "Invite" part of the event and Calendar fills that with the person's name. Right-clicking shows the email address has been found and is correct (it's a Gmail address). I click on "Send". Calendar doesn't seem to do anything. It doesn't say "Message sent" or "Invite sent" or provide any feedback to indicate an invite has been sent. The other person tells me they have received nothing. No invites, no emails with invites attached, no messages.

I checked everything Apple suggests. Apple's advice is not helpful: it says "If the email address you used to send the invitation isn’t listed ...." BUT, I can't find a way to control from which email account an invite is sent.

How exactly does Calendar send an "invite". Is it an email ? Is it an SMS ? Is it something else ? Just what should recipients get ? Is there a way to set the account from which invites are sent ?

Thanks.