First of all, a snowstorm is not “nothing more than a frozen rainstorm.” Frozen rain is just that, a raindrop that’s frozen from liquid water. A snowflake crystalizes directly from a vapor state into a solid, a process known as sublimation, and/or from very finely atomized water droplets in the air.
Second of all, snow storms/clouds can and do produce lightning. Rare is a winter here in Chicago when we do not experience at least one such episode. Infrequent yes, “no” lightning, no.
The process that generates the enormous release of electrical energy we call lightning is to this day not fully understood. The prevailing opinion is liquid water in the form of raindrops carries electrons from one portion of a cloud to another and results in electrical potential, but there are other forces in action man has not been able to properly describe.
However, lightning is definitely not caused by “too much water” in those clouds. Raindrops and snowflakes are a result of saturation, surely, but as you so thoughfully point out these do not always result in lighning production. (A much better description of the mechanics can be had at NASA’s lightning research website):
That being said, there is sometimes a fair amount of liquid water available in clouds that are generating snow but not liquid (or frozen) rain. The “electricity” is always there, it just needs an efficient mode of transport and snowflakes themselves may be lesser but not incapable carriers of electrons.