If you haven't seen it, here is the list of ways to get into download mode:
1. With the phone off and connected to the computer via USB cable, and battery installed, hold all three buttons, vol- vol+ pwr.
2. With the phone off and battery installed, hold vol- and vol+ (but not pwr) while plugging in the USB cable.
3. With the phone off and battery installed, hold vol- (but not vol+ or pwr) while plugging in the USB cable.
4. Attempt each of the above three options with the battery removed from the phone. The computer will supply the power to the phone.
5. Use Android Debug Bridge (adb) which will only work if USB debugging has been already turned on in Developer Options, since you can't boot into the system to turn it on. With the battery installed and the phone connected to the pc via USB cable, use the command "adb reboot download".
6. Use a USB jig. With the battery installed insert the USB jig into the USB connector of the phone. Some have reported that this method works when no other method would work.
It's strange that you can boot into recovery and not into download, so try all of these to see if you can get download mode some way.
I suppose it is possible that there is eMMC memory damage, but it would have to be only a very small segment of the memory for your symptoms. I'd be inclined to think the problem is not damaged memory, but I don't remember seeing those exact symptoms.