When using TfL single fare finder between a zone 1 station (say Kings cross) and orpington, applying a national railcard discount to this journey during peak time returns a price of £10.40. Using the forums ticketing service, and applying a 16-25 railcard, one obtains a paper ticket price of £8.85 during peak times. Am I missing something, or is it beneficial to be using paper tickets for this journey as opposed to oyster?
Railcard discounts work differently between Oyster PAYG and paper tickets (or digital equivalents thereof).
Paper tickets
For short journeys on the morning peak, most Railcards provide no discount on paper tickets. This is either because their conditions explicitly exclude use in the morning peak (e.g. the Annual Gold Card, or Network/Senior Railcard) or because they have a minimum fare in the morning peak that is more than the cost of a single fare.
A return fare might be expensive enough to meet the minimum fare rule though, and thus get at least a partial discount. And for the 16-25 and HM Forces Railcards the minimum fare doesn't apply during the months of July and August - this means it can be cheaper to use paper tickets than PAYG during those months, as you get the full discount on paper tickets at all times of day.
Oyster PAYG
With Oyster PAYG, all Railcards (except the Disabled Railcard) apply under the same circumstances: you get the discount for off-peak fares and off-peak caps.
On the one hand, this means that if you touch in before 06:30, you can get an off-peak fare which is also Railcard discounted - this would generally not be possible with paper tickets.
However, it also means that since peak PAYG fares are charged for most journeys if you touch in between 16:00 and 19:00, you may be better off using paper tickets for evening peak journeys - as you would not get any Railcard discount on the PAYG peak fare itself.
Confusingly, the evening peak fare would still count towards your daily off-peak cap (unlike pre-06:30 off-peak journeys), which is discounted. So there can be circumstances where you are still getting some (or full) benefit from your Railcard discount despite paying a peak fare.
For example I recently did a day trip to Heathrow on the Elizabeth line. I was charged a Railcard off-peak fare on the way out, and would have been charged an undiscounted peak fare on the way back as it was the evening peak. However, the daily Railcard Zones 1-6 off-peak cap meant I ended up paying only slightly more than the actual peak fare for my return journey. This meant it was still cheaper to do it this way than with a Railcard paper ticket.
Summary
Basically the long and short of it is that the cheapest fare depends on exactly when you're travelling, what kind of Railcard you have, and whether you're returning (and if so, whether the same day or not).
For Railcard holders, there isn't an easy, blanket rule of "PAYG is always cheaper" or "paper tickets are always cheaper". It depends on the circumstances.