Okay, I think I have an answer. I spoke to a Surface tech support rep who said that the Surface should definitely be able to run an external USB drive whether or not it is plugged in. Then he said, 'unless you have that turned off in your power options'.
So I went to look and it appears that there is a power option that causes this, and that the power option may be set differently while on battery and plugged-in. That would explain the different results we have been getting. Anyway, I changed the following
settings and now my Surface Pro will read my external USB 3.0 HDD.
Go to Control Panel/System and Security/Power Options/Edit Plan Settings. Then choose 'Change Advanced Power Settings'. A traditional old-style dialog will come up with one tab that says 'Advanced Settings'. Just above the drop-down dialog that shows the name
of the power plan you are editing (Balanced, in my case), is a link that says 'Change settings that are currently unavailable'. Click on that so that all options show in the window below. Now find USB Settings and expand it. There should be one option called
'USB selective suspend setting'. On my system, both battery and plugged-in were enabled. I disabled them and I can now use my external hard drive.
So here's the caveat. Yesterday, I couldn't get it to recognize any drive. I changed this setting today and now it recognizes my drives. I wanted to verify that this is what made the difference, so I turned them back to enable, but it still recognizes the drives.
This leaves me uncertain whether this is actually what made the difference. It doesn't rule it out. In fact, it would explain why different people get different results both with and without being plugged in. If the setting is just enabling the Surface to
suspend the USB power when it deems fit, then it may just be that it's not suspending it right now because I was just using it. I'm going to leave these disabled and see if the problem returns.
If anyone else is having this problem, I would be interested in whether changing this setting makes a difference.