I wonder if there is a way to modify the account's data without the program if you know its private key?
No. There are two concepts that are at play. Ownership and signing authority.
Only the owner of an account can debit lamports or modify its data. You have the private key, which gives you the signing authority of that account, but you're not the owner. More on ownershipaccounts herehere
The private key gives you the signing authority, which is required for any write instruction to an account. Even with this signing authority only the owner (a program) can actually write data to the account.