... is there some movement toward "upload the strategy and let the exchange run it"? which would provide a more level playing field, reduce energy and hardware costs, etc?
You mean enabling arbitrary code execution from a third party when you can lose billions of people's money in half a second? Also if two people want to make the same trade, who gets it? Exchanges do provide very limited special conditional execution instructions such as peg orders or stop orders, but it seems like a hard problem for them to support anything more sophisticated and general.
I agree with your sentiment but I would clarify that historically some exchanges did indeed create custom order types for larger clients that were not always public knowledge. I think that has mostly been eliminated but there are still a range of unique order types you can utilize depending on the exchange.