Using classic Outlook for Windows in business environments
Hi. I had the same problem in Outlook Classic and found a solution.
All the settings were correct, server, etc. However, when a message was written it would stay in the 'Outbox' until Outlook was closed and reopened, then it would send. It would not show as sent in Outlook (and yes, I had the "Don't save copies of sent items" unselected), but it did show as sent when I opened Gmail online.
When I did the suggestion in 6. above of setting the root folder path to INBOX the problems vanished! However, at the same time so did the 40 odd sub-folders that I had created with emails in a folder called 'Personal'.
So I removed the INBOX setting, the folders reappeared, and so did the sending email problem, as well as not showing sent items.
The problem was thus somewhow related to having all these subfolders. So I created a separate email account, moved all the subfolders to that, and had a 'clean' Gmail install. I reset the INBOX option on, and since then no issues.