Best of all is the system where they ring you back when they are ready
I have almost never encountered this while on hold (the closest I have come is the despicable "we are experiencing unusually long hold times", all the time), and it would be so good. What I particularly hate about the repeated announcements of the type you describe is that they are often much louder than the background music, so, not only do they train me not to pay attention, they force me to turn the volume way down, or hold the phone far from my ear, making it still harder to hear when a human comes on the line.
The trouble is, this was also their only support channel. No phone or web or email.
This went on for nearly 1 week when I was trying to talk to them to unblock my blocked-for-no-reason account a few years ago, and their response times got progressively slower.
In the end a quick message on social media got the account unblocked in 20 minutes. How?!