Is it possible to add memo checker to an XLM address so sending a deposit without memo to this address wouldn't be possible at all? This feature exists for BNB, XRP, etc.
You might be looking for SEP-0029: Account Memo Requirements.
SEP-0029 a standard way to define transaction memo requirements for incoming payments.
To set up memo prerequisites for an account, the owner submits a transaction with a MANAGE_DATA operation that adds a data entry with the name config.memo_required (in lower case following SEP-18 notation) and value 1 (a single character string "1" - ASCII code 49 or Unicode U+0031).