Anonymity preserving transactions on Ethereum blockchain?

Let's say I had KYC-free Ethereum so it was anonymous enough to begin with and now I wanted to transfer it to a wallet but I don't send it to (and eventually from) the same address each time but rather a new address each time for privacy reasons.
1. This is because I'm familiar with how this is easy and necessary for privacy on the BTC network, is it necessary on the Ethereum network?
2. What desktop wallets can do this if it is neccessary?
3. As far as I can tell the simpler (maybe a bit 'quick and dirty') way to achieve this is by just making a new account with just about any (desktop or browser) wallet software.

Yes I know hot/software wallets are a security risk etc etc. I need a hot wallet - the crypto won't be sitting in this account.
Oh and also does any non-custodial wallet, one where you are actually in control of the crypto have to download and save the entire blockchain as Geth does or is that just because that software allows node functionality? (Non-custodial Bitcoin wallets 'load the blockchain' but not in such a way that requires 100s of GBs of storage).

Thanks.