I'm not sure what type of hardware you're looking to use (i.e. a computer or a smartphone), but the only wallet that will provide you with the ability to do what you're requesting is Bitcoin Core.  I know, I know, 550GB to download the blockchain, but that's the point.  It's not only providing you with the privacy you want, but it's also supporting the blockchain and keeping bitcoin decentralized.  Obviously it's not going to work on a mobile device, but with your own full node running there are ways to connect your mobile wallets to it as well.

I don't know of any purely private wallets that can only download the blocks that are needed by your wallet (a pruned Bitcoin Core will actually do that, but only if the wallet.dat file was backed up within the last few days of opening it in the pruned version of core.)

Unfortunately, without an unpruned version of Bitcoin Core running locally, you're options (and privacy,) are limited.  Since you mentioned Sparrow, it can be used without a SPV server.  You can point it directly to an unpruned Bitcoin Core node running locally.