Do you have to hold liquid ETH2 tokens in a certain wallet to get staking rewards?

There are a billion liquid ETH2 tokens out there, but from the US I can't access Binance's or Kraken's. I'm also more interested in decentralized pools namely Lido and Rocketpool with their stETH and rETH respectively. From what I understand, you surrender your ETH to their pools, receive the liquid token in return, and then every day its quantity is rebalanced in your wallet to reflect staking rewards without any transaction (I still don't understand how one can add coins to a cold wallet without a transaction so please shed some light if you do). Further, you can purchase stETH or rETH from a liquidity pool for a slightly cheaper price. This bypasses Lido or Rocketpool's swap, but to the same effect.

My question arises in if there's an extra step to ensure your tokens are properly rebalanced since you never went through Lido/Rocketpool in the first place. I would assume they just scour the Blockchain to know who has keys to each token and rebalance that way, in which case it wouldn't matter in what wallet they're held. But maybe not, maybe it has to be in a certain compatible wallet, or maybe you need to connect it to Lido/Rocketpool regardless. Does anyone here know how the actual balancing process works?

Additionally, I've found you can trade stETH on Gate.io which AFAIK is the only liquid decentralized ETH2 stake token available on a CEX in the US (wouldn't it be nice if I could just use BETH or ETH2.S). The big advantage of this is that I don't have to pay enormous gas fees to stake, move it around, and trade in liquidity pools. But, if by doing so negates the auto-rebalancing protocol of Lido, then staking rewards are effectively lost and it defeats the purpose of the token. So will holding stETH in the custodial wallet of Gate.io still give me access to staking rewards, or do I need to move it to MetaMask/Trust/whatever and do I need to go through an additional step to guarantee proper rebalancing?