The general answer is no, but it comes down to whether you can trust owner of that website and any third party they use. And depending on how much privacy you want, Bitcoin Core or lightweight wallet which use BIP 157 with Tor or VPN are your only option. If you ask details about mutiples address or master public key to single block explorer or server, they might assume those belong to same people.
Using SPV wallet (lightweight wallet) like Electrum wallet, without Tor and connect it to other servers to use and check balance is not good for privacy.
It's explain why professional people are favorite to use Bitcoin Core, full node, Tor to do their Bitcoin transactions.
Use Electrum wallet with Tor.
Nothing is better for privacy than running a full node and checking the data by yourself...
I agree with you that using third party data is not the best.
Thank you to have you here and no one can disagree about risk from third-party tools but I don't imply you do something shady with your tool.
This website is exactly the same as using mempool.space block explorer.
When you add, for example, 3 addresses to bitcoindata.science, the website makes a request for each address in mempool.space API.
So, you have the same privacy level of checking 3 address separately in mempool.space
Looks a great explanation that helps me to understand the Balance checking tool from Bitcoindata.science better.