Is high gm, high current desirable for low noise?

At your risk.

Experience from Pleiades amplifier designs brings out that very small currents of the order of 50 to 100 microamperes give very low noise on electron tubes or JFETs. 

An explanation seems to be that at low voltage, (hence low current), secondary electron emission is minimum on electron tubes, see much earlier posts.

For example the Pleiades V6 battery powered electron tube microphone booster. The Pleiades DC90 RIAA preamplifier. The Pleiades 3 K117 RIAA, no hiss can be heard even with volume control at maximum. The Pleiades K117, 1.3V microphone pre preamplifier.

No wander this approach is also used on classic condenser microphones.

Having said this most of the devices chosen for these Pleiades low noise preamplifier experiments are high gm such as EF183 or K117. But they are used at low current hence low gm.