We have 4 options I can think of... I can talk to Enterprise support via the normal channel, I can talk to our SAs / TAMs like any customer who has engaged them, we can try and engage Pro Serve (like any customer), or what you're saying, I just ping the service team oncall, manager, product, etc on slack, ticket, email, etc. Only the 4th option is really different to external, and I know folks from 3rd parties with that level of access as well.

AWS isn't free for internal teams, we're still paying real money and that moves the KPIs.

Not sure what you mean by internal support, I was talking about the enterprise support or the team itself.

When I say fix for urgent things it's usually poking a management api, restarting some process, swapping hosts, etc. Or calling out some other issue. Same thing that enterprise support does (has access to) or asks the team oncall to do for any customer. Common obvious example is checking elasticache logs for odd error conditions or networking issues. Or checking if your instance is on a possibly degraded host/rack etc.

When I say fix non-urgent things, I'm patient. AWS has fixed many things for me esp if you look at the 1, 3, 5 year time horizon. I've got lots of problems, having someone solve them helps, many things can be hobbled along for 1-2 years if you know it's going to be fixed and you can fix something else. Much of my job involves paying attention to roadmaps for aws teams internal and external and guiding my platforms work to best leverage that.