This situation is a bit special because TfL charges have their own statutory regime. Drivers don't have to pay the congestion charge because of a contract with TfL, but because there is specific legislation giving them a duty to do so, and governing the mechanics to follow. All of that means that the situation is quite different from one where a payment is due under a contract and the parties have a dispute about its circumstances.
The congestion charge system is set out in various orders made under the Greater London Authority Act 1999 (section 295 and Schedule 23). Paragraph 6(8) of the order says:
Where a licence is purchased otherwise than in cash and payment is not received by
Transport for London (whether because a cheque is dishonoured, a direct debit, credit card or
debit card payment is declined, or otherwise), the charge to which the licence relates shall be
treated as not paid and the licence shall be void.
In paragraph 6A(8) and (9) there are some further rules about automatic payment on a saved card or by direct debit, including that
Where payment under paragraph (8) is declined for any reason—
(a) Transport for London may accept payment by any other means it considers suitable
in the particular circumstances of the case; and
(b) where all outstanding charges under paragraph (8) are not paid within such period
as Transport for London may specify Transport for London may suspend or cancel
the CC Auto Pay Account to which those charges relate.
The overall scheme is one of strict liability for the keeper of the vehicle, but with various options and time periods for the payment of the resulting charges. As written, TfL has several opportunities for discretion, e.g. it can choose not to issue a Penalty Charge Notice, but the keeper has little choice except to make the payment by one of the available means. In the situation as described, it's common ground that (1) a payment should have been made, and (2) it wasn't made. Regardless of whose fault (2) was, it's still the keeper's duty to make the payment. Disputes on this point are routinely knocked back by the tribunal that deals with such matters. They typically say that the keeper ought to try to make the payment by some other means, within the time allowed.
In the cited instance the keeper (with pressure from the press) was able to convince TfL to use their discretion. If that failed then the statutory route gives the keeper some opportunities to contest the matter, but unfortunately for them only on certain specific grounds - the tribunal path gives no discretion to adjudicators to do anything other than follow the strict process. This was the subject of an appeal, Walmsley v TfL [2005] EWCA Civ 1540, in which the claimant accidentally paid the congestion charge using the wrong vehicle registration number, and TfL pursued her for not having paid for the vehicle which she actually drove. An adjudicator said that she had to pay, applying the regulations as written. On review in the High Court, the judge agreed that she had to pay, but said that the adjudicator had general discretion to cancel a PCN and could have done so in this case. But in the Court of Appeal, a three-judge panel said that the adjudicator lacked that power, and that someone in the claimant's position would simply have to hope that TfL decided not to escalate the matter.
The judges did leave open the question of whether TfL had properly exercised their discretion on that point, as that hadn't been explicitly challenged. Obiter, Lord Justice Sedley criticised the setup where drivers whose grounds for contesting a PCN aren't foreseen in the statutory structure, are limited to hoping that TfL shows benevolence:
For it is unfair that those who, despite the absence of any indication that they can do so, write to TFL in the hope of clemency, at present obtain an advantage over those who assume, from looking at the Regulations, the penalty charge notice, the appeal form and TFL's website, that there is no way of doing any such thing, and pay a fine which they ought not in fairness to be required to pay.
Whether it would be better, instead of TFL's acting, however conscientiously, as judge in its own cause, for the jurisdiction of the Adjudicators to be enlarged to include a power to remit fines for non-scheduled reasons is something which the Lord Chancellor, who is the Minister responsible for making these regulations, might want to consider. For the present, however, it is not the law.
...and it is still not the law in 2024.