I've owned or ridden quite a few automatic shifting bikes. They all have issues.
CSA Autobike. 3 sizeable weights in the rear wheel linked to a moveable spoke protector that pushes a reverse sprung derailleur as you go faster. Plastic plungers in the linkage that are very prone to fouling with dirt, grit, water. Not useable when dirty. The CSA Autobike itself was a super cheap pile of junk. You have to ride at specific speeds otherwise it's in between gears and trying to shift. Lots of grinding, skipping, etc. You had to slow way down prior to stopping to give it time to shift back into low gear so it was ready to go again without trying to shift while starting up. You basically had 5 speeds you could ride at, too fast or too slow and it would grind away. Owned one, sold it, got another.
Huffy Autobike. Same rear end on a different cheap bike. currently have one.
Landrider. A mini timing belt driven flywheel directly on the derailleur. Much more compact than the Autobike , a decent quality bike, but way overpriced. The shifting parts probably cost as much to produce as the rest of the bike. It has the same speed problems as the Autobike. It was adjustable overall. One I have shifts at reasonable speeds, the other, well you have to get up to about 15 mph go get out of low gear despite the adjustment sett to the minimum. It has a standard double ring crank and front derailleur but that was mostly useless at normal speeds. Landrider probably spent more money on tv advertising than any bike company ever. They ran countless 30 minute infomericals in the 1990s. . Have 2 in stock.
the Autobike and Landrider required you to ride a constant speed up hills. It would try to shift under load if you slowed down. You would get some warning as it would start grinding gears. You had to slow down at the base of a hill to what speed you thought you could maintain and ride that speed all the way up the hill.
Trek Lime. A overly stylish bike with a generator front hub, hidden wiring to a servo box under the chain stays that pulls a cable connected to a 3 Shimano rear hub with coaster brake. Shifted quite well. A secret switch for different modes: no shift, lax, mid, aggressive shifting, hidden in the servo box. The main problem was the bike itself. Far too much spent on the style and a lousy ergonomic position. Too low production numbers to get the cost down. One issue was going up hills, you had to ride the same speed all the way up a hill otherwise it would quickly shift under load without warning. The only bike I've seen with a storage compartment in the fat seat. Had one sold it.
With the Autobike, the Landrider and the Trek Lime, a digital speedometer would be a good idea, then record the specific speeds where it's happy, and write those numbers on the handlebars.
Shimano Auto-D demo bikes. Shimano developed a electric shifting 3 or 4 speed system and sent kits to many bike manufacturers so they could build bikes around them. Shimano took the bikes to dealer shows to demo. Only a couple companies actually put Auto-D bikes in to production for some very high prices. Mainly urban commuter bikes. The demos I rode were very nice. I liked the soft touch shifter controls better than the auto mode. This was before the Trek Lime that had no handlebar controls.
The devices on ebay will have even worse shifting problems than the Autobike or Landrider which shift based on speed alone. With it responding to chain tension, each pedal stroke could have it trying to shift. For all you who ride in too high of a gear ratio, it will down shift when you don't want it to. Any attempt to stand up and pedal hard will probably throw you over the handle bars when it shifts to low unexpectedly.