If not, I would like to know if there is a way to check the balance of an address using the btc network and not a blockchain explorer, for example with some node calls running btc core.
You can create a watch-only wallet in electrum just using your address.
If you want anonymity while using blockchain explorer, you can use Tor browser to check the address or addresses that you want which is better than using Electrum that connects using IP address, or enable Tor on Electrum for it.
For privacy, you have to run your own node, for not to depend on central server.
Just keep in mind that whenever you use a central server, your IP may be stored. Another option is to stick with private coins like Monero or Pandora Cash that come with default anonymity. By doing so, whenever you use the blockchain, you always have plausible denial.