The administration and maintenance of Microsoft Exchange Server to ensure secure, reliable, and efficient email and collaboration services across an organization.
Kindly note that the tag "Exchange Server Management" is used for Exchange on-premises server questions.
Since you are using Gmail API on your VPS, which we are not very familiar with, if possible please consider contacting Gmail support and your VPS vendor for better support.
If "Outlook" in your post refers to an Exchange Online account, and you also have admin access to that organization tenant, from Exchange Online side you can use message trace to check if the message actually reached Exchange Online server, or possibly get quarantined or blocked by Exchange Online Protection.
More detailed information: Message trace in the modern Exchange admin center in Exchange Online
Besides, since sending from local PC the email can be received by Outlook, the possible cause may be the VPS's ip address is on some blacklists or blocked by Outlook servers.
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