They're probably going to wonder why they are wasting their time with me when I can't even get something "as simple as email" right.
For personal email I'd probably consider it.
AFAIK there is some interface in business gsuite, available only to domain admin and not regular users, where a limited subset of the above can be done.
The same applies for inbound email too -- they claim they are trying to send me an email, or getting an email from something like a network printer fails. Good luck trying to find inbound email logs for the entire gmail.
Besides, there's a bounce email that describes errors.
It's their system that is misclassifying their email, not you being on a "hipster" email system, whatever that means. At no point is it your system's fault so don't even begin to start phrasing it as such.
From your perspective, your email was delivered successfully.
But to the recipient, you claim to have sent an email that is nowhere to be found in their inbox or spam folder, and if they contacted IT (keep in mind that 90% of people won't even bother doing this much), they'd say it was never received - why should they believe you?