I've only staked a worthless altcoin, where staking worked straight from the Bitcoin Core clone, without exchange. I assume the same would work on ethereum, but the minimum is about half a Lambo worth of that coin. So staking leads to further centralization, who would have thought.
It's a bit different with Ethereum in that (a) you have to send the coins the the beacon chain and while you can hold the keys, you can't really withdraw the coins until some future hardfork and (b) there are uptime requirements - you can get penalized if your staking node goes down. Looks like this Ledger service is trying to solve these two issues (not sure about the first part though but presumably you can sell the staking tokens, perhaps at a slight loss), as well as allow staking less than 32 ETH. Nothing particularly wrong with that, there are other similar services, but it's definitely perpendicular to the purpose of a hardware wallet.