Plantronics was not able to help and as far as I can tell not really willing to investigate to find the root cause. They basically say they can't do anything anyway because there is no way to update firmware on the BT earpiece. The tech support person acknowledged that they have had multiple reports of this, but Plantronics thinks it is on Apple because it works on previos iOS/iPhone versions as well as Android. I pointed out to them that I have other BT devices that stay connected. I also pointed them to a Bluetooth "sniffer" product that can sniff packets between BT devices, but again no interest on their part. They also told me that the one that I have (date code Feb 2018) is the same as what is shipping today. That means there is no update and if you buy the M165 today you could have the same issue.
That being said, I have a workaround and an alternative for anyone interested. The workaround is that I found on iTunes a completely silent music track that you can download for $0.99. It's called "A a a a a Very Good Song (Silent Track)" by Samir Mezrahi, and it is just 10 minutes of silence. I created a playlist that plays it numerous times, giving me enough for about 3 hours. I can play that when I want the M165 to stay connected but not listen to music, like when I am driving and want to hear GPS directions or I'm waiting for a call.
The alternative is that I found a pretty inexpensive BT earpiece from a company called New Bee. The model number is LC-B41, on Amazon and with tax it was less than $19. I've had it for a couple of days and it doesn't drop. It's a little bit heavier and in some cases the sound quality is not as good but it's not bad at all - for under $20.