"I advised that only $7k per year conversion allowed for Roth"

Whoever told you that does not know the rules and should be corrected by you, or you should start using a different mentor. There is no limit for conversions. Limits applies to Contributions.

"Both he and his broker insist that there is no taxable event here."

The magic for a Backdoor Roth to work, is because any funds in the account(s) of that same type are Basis (never tax deducted and no earnings). Otherwise, any amount converted (which really is all you are doing with backdoor) is taxable on a pro rata basis. If your taxpayer had put all the money into the Trad IRA at once and immediately converted it to Roth, and has nothing else in any similar tax-deferred type of account (as per the listed account types on the Form 8606), then there would be no taxable event.

Your taxpayer had a delay in the contributions. If there are any earnings against the earlier amount, or against the total prior to the conversion, then these amounts are the taxable amounts, as pro rata to the amount converted.

"I advised that only $7k per year conversion allowed for Roth," <== not correct

"so the additional $7k is taxable as a regular distribution." <== applies anyway, as a Conversion is always a Distribution. The issue is how the Distribution is treated. A Backdoor Roth = an immediate distribution as direct transfer, at the time of contribution and conversion being the same date, to avoid having taxable earnings and to avoid triggering any required withholding.

"In addition, he over contributed to the 2021 Roth and incurred penalty."

That can be dealt with by a Corrective Distribution. You would then report and pay excise tax of 6% on those earnings and do not roll/convert that amount.