Is it harder for a low-battery DMM to measure current in a comparison to measuring voltage and resistance?
I have several DMMs, and most of them measure 0 on the current range because I'm clumsy and blow the current range protection fuse, then don't get around to replacing it because it's an oddball size.

Of course it's possible that you've toasted other components(*1) used for the current input. I'm never very sure whether the fuse provides complete input protection, especially in cheap meters.

Try replacing the fuse. And if you only have one meter, buy another couple of cheap ones, and dedicate one to current measurement. You'll often find situations where you want to measure several things simultaneously, and it's comforting to have a spare.

*1) The favourite to look at is the shunt resistor. Its power rating is limited so you can burn it quite easily. That's why all these warnings are printed on the DMM. ;-)