I mean, here's one instance where gmail's uber sophisticated AI seems to have classified a seemingly legit email spam?

For a company that prides itself in customer service, I don't see any other way but to run their own email exchange (ala Amazon) because their private systems can then discern between email-ids of paying customers and spam, if nothing else.

I mean there are things about Gmail that might make one think - I shouldn't use this as the email of my business if it is important to me - lots of them exposed and upvoted on this forum.

I've run my own for 15 years with no problems. So I recommend learning how to set up postfix. If you prefer not to do that, people here seem to praise fastmail pretty uniformly.

For a small startup, I like GrooveHQ. It’s like the Zendesk experience but less janky, much simpler, and well suited to a small team. Just have your support emails forward to Groove. Will still be email support from your user’s perspective. You can keep your current workflow going but turn on forwarding to try it out.

In my experience, any ticketing system will be a huge improvement over using a shared gmail inbox for support.