Helical bevel gears including hypoids have even greater lubrication needs than right angle spur gears. They're a hybrid of worm gears and spur gears. The gear faces actually slide against one another like worm gearing. That helical gearbox in the photo might be fine for a hobby helicopter, to fly it for entertainment, but it won't last long in any sort of vehicle for transportation.
Even though I love to look to new ways of applying existing ideas, it usually helps to look very closely at the designers intentions and try as much as possible to respect them. Hand power-tools such as drills and angle grinders put a very high premium on power, portability, and cost savings. Two compromises made to achieve this are to use extremely high speed input (brushed motors that generate peak torque when screaming such as your vacuum cleaner) and by using gear reductions that, while robust enough for adequate power, are still classified for "intermittant" rather then "continuous" use.
A lot of industrial stuff is so rated, particularly motors; either capable of operating all day long in a factory, literally 24/7, or only for short bursts of no more than a couple of minutes at most. The former will, in the case of gearing, usually have extremely rigid support of shafts and bearings via cast iron housings providing minimal deflection in the bearing seats and from the EXACT intermesh of the gear faces as was originally designed, maximum damping of harmonics/vibration, and serviceable oil baths. Hand tools use light aluminum housings and thermally stable grease. The housings are generally ungasketed and sealed only enough to keep contamination out, and heavy grease in. Most hand tools are essentially disposable machines, despite the odd milwaukee hole borer your grandpa lovingly passed down to your dad. Motorcycle and automotive transmissions are classed somewhat in between.
Part of the history of machine lubrication is actually interesting. Well into the last century methods of mass production didn't yet exist, but engineers made pretty remarkable progress on one-off machinery none the less. It was prohibitively expensive to build something as extravagant as a sealed case around your entire gear, transmission, valve train, etc to hold an oil bath when after all you could just hire some guy (or even child) a nickle a day as an "oiler" to scamp about the ship or the factory all day with a can to keep everything lubed. Clambering over, under, and even within the spinning works, this is probably where the expression "grease monkey" originated (and is still a perjorative for the lowest paid guy in the shop). Even the earliest airplane engines had exposed valve trains. The pilot would oil up the works by hand, spin everything up, hope for the best, and end up with an oil shower. Flying was an extremely dirty job.
Sorry for that little bit of pure trivia. I wanted to be an engineer but flunked out of math classes (twice) in college. Had to change direction. Anyway. I think that shoehorning power tool transmissions into an e-bike is kind of asking for trouble. The gears and bearings were intended for intermittant use, and linear loading in most cases. Adding pulleys and sprockets introduces loads that the designers made no margin for.
