In my opinion, and I am not a tax expert, your father has a legal ownership interest in the home since you have added him to the deed. Since your father has a legal ownership interest and is actually paying the interest, he is entitled to deduct the interest on his taxes.
I assume, however, that your father may not be able to itemize deductions and is okay with letting you take the deduction. If this is the case, your father should gift the payments to you thereby making you the actual payer of interest. Your father would have to declare the amount gifted that is over the annual limit of $18,000 ($19,000 in 2025) to you on form 709. This doesn’t mean he has to pay a tax on the gift, it just goes against his lifelong gift limit of $13.6M.
You also need to consult a tax expert on the need to file gift tax form 709 of your own for gifting 50% of the fair market value of the home to your father by adding him to the deed.