tl;dr -- That penalty interest rate is essentially a legal technique to improve the lender's recovery rate in the event of a default. They lender uses it when they predict a borrower to be in financial distress already...and therefore highly unlikely to repay in full. I think this timing issue is the core of your question.
The answers from @CactusCake and @D Stanley to the original question you linked to both explain a risky interest rate as compensation for the risk of the borrower defaulting on the lender.
Extend that explanation by thinking of the interest rate on any form of credit (bond, bank loan, credit card, rent-to-own, etc.) as the lender's profit margin. To be profitable, the lender has to cover their costs. The lender's expected loss due to default is one such cost, and it typically accounts for a significant portion of the credit spread (that part of the interest rate in excess of "risk-free" interest rates). reference
If a borrower is 60 days past due then their probability of default is likely to be relatively high. Apart from the common sense element of the situation, lenders rely on credit transition matrixes, which estimate the probability that a borrower will default over some period of time given their starting probability of default. related reference
Since your question is hypothetical, assume the credit card company in question is located in the U.S. where non-mortgage lenders generally have a legal claim ("recourse") against borrowers in default. The cardholder's lending agreement most likely specifies the card company's right to apply that penalty interest rate -- it may well be as high a rate as can be applied without violating usury laws. By the time the cardholder has gotten this delinquent, the card company assumes the cardholder will default. The company then applies that penalty interest rate fully expecting that they will need to pursue a legal claim to recover the past due card balance and the money value of the penalty interest rate (which they assume has a high probability of similarly going unpaid).
Further expecting that the cardholder will not have the resources to pay off the card balance in full and that the risk of cardholder bankruptcy is high, the card company's penalty interest rate is effectively boosting their legal claim from, say, $1000 to $1000 * [1 + (.2999/365 * days outstanding)]. In the U.S. -- where we assume this all takes place -- creditors with similar claims (credit card, utilities, car loan lender, etc.) are generally entitled to a pro rata share of a bankrupt person's assets based on the size of their claim.