'Conventional' mechanical transmission from turboshaft to propeller includes losses in the gearbox and whatever intermediary t-boxes. For the layout shown i would be surprised if you lost 6-5%.
For a hybrid-electric aircraft, the turboshaft spins a gearbox (95-98% efficient), which spins generators (95%), which use cables with non-negligible heating losses (99%), to power motors (95%) of a certain efficiency. As you can see there are more links in the power chain and more losses.
I believe where hybrid-electric shines is it enables things like distributed propulsion, redundancy, flight controls that rely on thrust and not control surface deflection, and other nifty tricks that are second-order effects, so you need to take advantage of a bunch of those to claim superiority over a mechanical transmission.
For a purely electric solution, you are right that battery specific energy is the key enabler. However i'm a bit more pessimistic on the timeline. For man-rated commercial flight, you will need mature batteries (not laboratory experiments) that can be manufactured in large volumes and are affordable. Right now i would peg the limit at 250 kW.h/kg (things you can buy in reasonable quantities at reasonable cost). Who knows, if Elon really can deliver on his mega-factory he just might make the goal closer...