TLDR Eventually I was able to get the Nintendo Pro Controller to reconnect fine using an older USB Bluetooth 4.0 dongle on Windows 10, specifically the ASUS USB-BT400. A newer BT5.0 (ASUS USB-BT500) didn't work for me, nor did a TP-Link UB400 (USB 4.0). Now it reconnects without having to re-pair it each time.
For those that find this thread from a search years later, I was having the same issue with my Nintendo Pro Controllers when attempting to reconnect them to my PC running Windows 10 via Bluetooth using a USB to Bluetooth 5.0 dongle (ASUS USB-BT500). They would pair just fine, but if Bluetooth was disabled/reenabled or the controller was turned off and then back on, the controller wouldn't reconnect. It would require a re-pair to start using the controller again, similar to what
@mangaTom mentioned.
Upon an attempted reconnect, the "Bluetooth & other devices" window would show the "Pro Controller" still "Paired" and then it would say "Connected" upon clicking one of the buttons (A for example), but the controller's front lights would go back and forth without connecting like it was trying to pair. Eventually the front green lights on the controller would turn off, and then Windows would go from "Connected" to "Paired" again on the same "Bluetooth & other devices" settings window.
Out in the Internet ether, others mentioned they had success when using a Bluetooth 4.0 dongle, so I bought a TP-Link (UB400) to give it a go. However, I was seeing the same reconnection behavior from the Pro Controller with the US400, in that it would pair and work just fine the first time around, but it wouldn't reconnect after being disconnected. The UB400 didn't work either.
A solution that worked for me is trying an ASUS USB-BT400 dongle I had around that wasn't being used. I first removed the device from Windows on the "Bluetooth & other devices" window with the UB400 connected - Windows for some reason pairs a Bluetooth device to the dongle itself instead of the Windows instance - then re-paired the Pro Controller using the "new" USB-BT400 dongle after removing the UB400.
The pair was successful and it now reconnects fine without having to re-pair it each time I want to use it.