"and then essentially deposit the same funds back into your IRA as a 2021 deductible contribution."

Don't confuse distribution, rollover and transfer. A contribution has absolutely no bearing on the source of funds.

Qualifying to make that deduction and qualifying to make an IRA contribution, are each/both subject to yet other separate things that have no bearing on the source of the funds you intend to use for that contribution.

You have to have earned (wage or working) income to be able to make a contribution, and if you cannot afford to fund that contribution from funds on hand, you then could have sold a kidney, drugs, or taken the RMD as the source of the funds to deposit. That RMD doesn't affect its taxability the way a transfer or rollover might. The RMD is a type of taxable cash inflow, no matter how it gets used.

There also is no direct correlation between the RMD withholding and the Taxes owed. Each item stands separately. The RMD is taxable; period. The deduction for the IRA contribution might or might not wash away the same amount as the RMD creates. Their 1040 has more than just these two activities, and must have earned income at the least, as a third activity. The withholding on the RMD is nothing more than a prepayment of projected tax liability, the same as wage withholding isn't tax on wages; it's a computed projection on the entire 1040 collected through wages, if the TP chooses this method.

You can always choose to have no withholding and pay quarterly estimates.