We are talking about two different applications!
Yes, grease the heck out of your coaster hub. The first couple times you brake, you will not see much power at all. But after that it will be fine. The grooves vent grease enough to let the brake system combine both high pressure against a revolving surface with fluid resistance (the very thin film of grease there now).
There is no way (without redesigning coaster hubs from scratch) to prevent this contamination of the brake shoes, except for running the hub without grease.
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The external band clamp brake, such as Grubee HD hub, is probably what you are looking for. It requires a lever to operate, not foot pressure.
re: grease coaster hub. Think of it this way: On a 1" threaded headset, most people lightly grease the inside of the steerer to prevent rust/siezing. Try to turn that sucker (with that hardened ferrous steel grooved surface pressing into the tube, the stem nut tight). Not turning easily. But it will turn if you smack your grip on a stopsign or tree, or dump the bike over. Thats a lot of friction for a tiny tube diameter and one "shoe", right?
Note: If your coaster hub brake starts degrading in performance over time, due to normal (well accelerated with motors) wear, you can possibly buy replacement shoes, or hacksaw new grooves. Also when you reassemble the hub and re adjust tightness of the hub (cones and locknuts), you will improve the reaction of the brake system to the movement of the pedals opposite of normal pedalling direction.
Drum brakes on a car, or disc caliper car brakes, both are designed for stopping something operating at thousands of times more a demand for mechanical work (friction/stopping power). They have their shoes isolated from greasy bearings, but are subject to contamination. Any grease, gas, oil, or brake fluid on the pad or rotor (one leads to the other instantly) would require a laborious (expensive?) change of parts.
The bike coaster hub is somewhat sealed in its own happy greasy environment. Nearly zero maintenance! Enjoy!
Ps leaning back on the bars to shift your weight over the rear wheel while standing or hovering over the seat will likely aid in leverage to activate the hub brake, esp a sudden but smooth motion, like a BMX'er about to do a manual over something.