Is there an ideal (ish) cartridge output?

Getting a quiet result from a MC phono stage (or MM stage with SUT) and ~ 0.5mV cartridge is fairly easy to do. It gets increasingly difficult as your cartridge output falls below that. At 0.3mV you’re usually dealing with some discernible noise (unless you only listen at modest volumes), and it’s up to you whether that’s going to be problematic or not - though with good setups it should not be notable with needle in groove. Of course this is assuming you’ve optimized grounding so hum isn’t an overriding issue. And I don’t have much experience with MM cartridges.

Cartridge output along with phono preamp gain setting and making sure you are not inputting more than your preamps input sensitivity (mV). Risking not so much clipping as simply raising the noise floor too much, most preamp sections can handle close to 1V but some are spec’d at like 300mV. So some extended listening is needed.....Clearly a lower output in the 300-400mV range will help keep the noise floor pretty low, while allowing all the subtle musical pcs to come thru.

what is the best output to give you the lowest noise without sacrificing performance. 
Higher output is definitely better, and would be universal, except its not free. There are tradeoffs.  

There are only three ways of getting higher output: more coils, bigger magnets, or more powerful magnets. There's a few more little tricks but basically that's that. Expensive materials basically forces everyone to using the same stuff at the same price level, and so it then comes down to how much. How many coils, how big the magnets. 

Either way, all things being equal, higher output equals higher moving mass - equals less detailed tracking. This is why at the top of the game pretty much all the best MC carts have pretty low output. Its simply not possible to track as well with more moving mass as less.

This also explains why Strain Gauge works so well. With a fraction the moving mass but much greater output (not to mention no RIAA EQ required) its a technological no-brainer.

The so called ''moving parts'' by an (MC) are cantilever/stylus
and coils. Those are fastened to the joint pipe in which also
''tension wire'' is fastened in order to connect moving parts
with the generator. It is obvious(?) that cantilever /stylus
combo weights ''nothing''. So the most weigth is in the
coils. The number of windings determine the output. So
the lower the output the lower number of windings and
the lighter the moving parts. Any attempt to reduce the
weight of the moving part means reduction of the wire jn
 the coils. One can increase the output by ''iron core''
in the coil former but this has other drawback. The ''coreless''
coils are the best but with the owest output. For those
one will need very good phono-pre or SUT.

The most noticeable noise I get when replaying vinyl is from the record surface, the phonostage noise doesn’t seem to intrude. I looked up the specs for my phonostage and it claims a SNR of 82dB for a 0.5mV output and a gain of 64dB. I’m actually using an SPU Royal N with a 0.2mV output. (I have a VPI 16.5 and a Degritter to clean the records which helps).

If you’re after a medium output the EMT cartridges with around 1 mV seem to have a good reputation for not losing the music in the hifi.