On the off chance this helps anyone - I was having the same problem but it turned out it was because I'd created a rule in Outlook (Office365) to delete messages from "App Store Connect", as at one point I was receiving a lot of noise when we had issues with our app store uploads. Test Flight invitations come from the same email address (noreply at email dot apple dot com) so these were automatically getting deleted too, and didn't show up even when I searched using the local Outlook client. Finally thought to try searching with the web client and sure enough, they were sitting in the Deleted folder.