I wouldn't care about the cheating part but if you are able to drive considerably more power through your back wheel than you could without the motor please stay off of mountain bike trails. The trails get destroyed by the back wheel spinning up coming out of every corner and you end up with heavily rutted motocross style erosion.

Depending on your locality and the power of your eBike, they may be right -- eBikes are restricted on many trails.

Yes, but in my area they are allowed on bike trails unless specifically signed otherwise (I've never seen such a sign).

I've never heard of these trail restrictions on the east coast. I wonder if it is a west coast thing.

It's really hard to generalize since state and local laws vary so much.

For example, in Massachusetts, eBikes are allowed on bike paths, but not off-road bike trails:

https://malegislature.gov/Laws/GeneralLaws/PartI/TitleXIV/Ch...

Motorized bicycles may be operated on bicycle lanes adjacent to the various ways, but shall be excluded from off-street recreational bicycle paths