The person initiating the interaction should make it as easy as possible for the other person.
For example I work with multiple
different timezones, I have meetings all morning. I can quickly answer questions if they’re straight to the point. Daily I’ll get 10-15 people
contacting me about X topic. If it’s not straight to the point I can’t help, it will be after lunch before I can and then it’s probably too late.
I think if "hi" is causing problems, you have larger process issues. For instance, it sounds like you're overloaded and understaffed. The issue isn't with "hi", it's with running the machine at too high a pace to absorb even basic social interaction without derailing. That's a problem.
Maybe a shift in perspective might help. It sounds like people need a lot of help from you. I'd help them and do it in their preferred communication style, building relationships in the process. Then I'd show management how much I was doing this and use my newly strengthened relationships with the team as a foundation to ask for a significant promotion. View people reaching out to you (even if you think the request is silly) as an opportunity.
For example I work with multiple different timezones, I have meetings all morning. I can quickly answer questions if they’re straight to the point. Daily I’ll get 10-15 people contacting me about X topic. If it’s not straight to the point I can’t help, it will be after lunch before I can and then it’s probably too late.