Thinner? Depends, sort of/sometimes.
Sharper is most important for clean cuts. Some folks swear by carbon steel for max level of sharpness capability, not that it matters if not sharpened properly. Real razor blades for shaving need to have, and the market ensures they do have, a VERY sharp edge for comfort. Customers just won't shave with a dull, painful blade. Single edged razors are more commonly used for craft purposes where sharpness isn't so critical, so less market reinforcement of sharpness.
For real light balsa, thinner becomes important to avoid damage to the wood edges BEHIND the cutting edge. But its harder to control because it can be floppy.
For med to high density balsa, you need a heavier blade to have control of the cut.
Note, the real advantage to carbon steel vice stainless double edge shaving razors is the ability to break and reshape the blade to more usable shapes. Carbon steel breaks with clean edges, stainless steel always bends and leaves a burr that snags in the cut.
Further note, there is a pretty good competitor to double edge razor blades for sharpness, real scalpel blades, not craft blades. Sharpness demanded by prime customer, surgeons, benefits us hobbiests. And they come in a nice variety of shapes and have nice handles, improving safety over double edge razors. Some of the previous links also sale real scalpel blades and handles. I just use my generic X-acto handle to hold them.