Yes, that is a nightmare. You know how many messages you need to send over chat in situations like this?
ONE. Not at least. AT MOST.
This isn't like yelling across the hallway and being unsure if your friend heard you.
This isn't like coming to your spouse's room to tell them dinner is ready, them not coming in 5 minutes, and you coming back to get them again.
Work chat systems show read receipts. YOu know if the person got the message. And there will be a permanent record for them to see it.
If they're not responding there are two possibilities:
A) They saw the notification/you message
B) They didn't.
If it's B, more messages won't help notifications appear.
If it's A....don't you maybe have some trust in your coworkers (presuming this isn't a 15 year old on their first job at MacDonalds) that they know how to prioritize and if they're not responding to another human being it's probably because they have some reason not to?
They might be mid conversation.
They might be dealing with a production issue.
They might have a screaming baby in the room next to them.
And in all these cases, if you give them a minute, they'll arrive, apologize profusely for missed the 1on1 or keeping you waiting, and either proceed or reschedule.
Meanwhile, you've already blocked this time off for the meeting. So go and reply to some emails, or work on a promo doc, or hell if you work too hard already, take a break, and click around on hacker news for a few minutes.
It'll be fine.
I would absolutely without hesitation would not work for, and would quit working for a manager who treated me with three messages like you just described.
"you there?"
"still waiting... ETA??"
What a nightmare control freak, huge red flag, jeez...