Private coin pegged to Public coin

Hey ya'll. I Would like to pick your brain on an idea and whether its feasible/useful or has a valid use case or just pointless. There are many stable coins with various ways they stay pegged to an underlying asset. I'd like to explore the option of creating a private coin, so on a private/permissioned blockchain, where the coin would be pegged to a public coin like bitcoin. Now why would I even want to do this instead of just buying bitcoin? There are a couple reasons I believe where I would want to do this but won't hit on that yet. Wrapped Bitcoin is Kind of similar, in a way, but its a public chain.

So here's my process, create a private centralized blockchain with a private coin that is pegged to Bitcoin. This coin has limited use and can't be bought or sold but can can be given out as a reward that can be used for/with/within the organization of the private network. This would be my base program but I would want to be able to have the capability to later on add in an exchange process (can now buy and sell for fiat) then also have the capability to transition it to a public blockchain (remove the peg then?), maybe by a type of forking process? I know this may seem odd but see a a plausible business use case that would fit to this concept. Could this be possible technically speaking? what are some flaws in my thinking? Other things I may not be considering? Or is this just stupid idea? Any comments or further ideas would be helpful in brainstorming on this idea. Thanks.