Don't think one is on any of my e-bikes, and probably wouldn't use one if I did, but they may make sense to an individual who needs to bring his bike up a hill too steep to ride.
Yes this is correct. I have "crawled" up ridiculously steep hills at like 2mph before using the throttle. I'm probably wrong but they felt like they were like > 60% grades. Any faster and I would have probably lost traction and slid down.
I don't know, it seems like a reasonable filter to me, if you can't push your 45lb ebike, you probably should only ride it places where you won't need to push it.
I was just describing what seems (to me) to be the function of the "walk" mode, not arguing for or against. I don't care whether throttles are allowed or not although, as I've said, some levels of PAS are easier to use than throttles IMO, FME.
Now look a little more at what is a walk assist button. Even with all the bad faith you put in it you should find a difference with what you name a throttle.