You can't download the chunks of blocks since they would be meaningless without having the blocks before it.
I don't think that is meaningless. Of course you have to compare the previous block hash included in the first block of your chunk with several sources (e.g. at all block explorers you know if you're paranoid

). But if you know e.g. that you received a Bitcoin transaction one day before, but want to check the correct amount without having to fire up a SPV wallet (which potentially may expose you to Electrum server privacy attacks) then such a service may be quite neat as you can check all blocks of the last 24 hours approximately. You do all the analysis on your own device, without exposing the addresses you own, and without needing a full node.
Also if you want to do some onchain analysis of certain blocks on your own (e.g. to look for hints which addresses could belong to a single wallet) and you know approximately the timeframe where you're searching, that would also be an use case for such a service.
In the end, that kind of service could basically enable everything a normal block explorer can (even what walletexplorer.com does, i.e. search for "connected" addresses which may belong to the same wallet), only that you need slightly more resources on your device, but far away from the resources you need for a full node.