If my retired mother gifts my brother a $10,000 disbursement from her 401K, and the 1099-R with this transaction has tax withheld, will she be the one paying taxes?

She proceeded with this $10,000 gift because she was told by another family member that neither she nor my brother would be taxed for this gift amount, and because the amount was less than $14,000 there wouldn't be a gift tax either. However, I suspect that this family member wasn't aware that my mother's retirement fund is in a 401K and not an IRA, so all the literature I've seen appears to note that disbursements have to be taxed regardless. I'm looking to seek clarity on the matter before proceeding. Thank you.