If you turn up with £30, a completed application form and (where applicable) proof of age and a passport sized photograph you will be able to buy any of 16-25, Senior, Family & Friends and Two Together Railcards immediately. No need to have any previous proof of purchase, no need to wait for a week.
You seem not to be seeing the point I'm making. Which is that, again
logically it isn't fair to ask people to buy a new card when they already have a valid card and either have forgotten it on that occasion or are entitled to a replacement at reduced cost in the event of loss/theft.
For example, say a railcard is stolen whilst away on a trip and the ticket has a value of less than £90. It isn't necessarily possible to get a railcard delivered locally if the original was purchased online, and you can't get a replacement if you purchased it at a station because the receipt is kept safe at home. Only if the ticket is worth £60 or more (and thus the difference between paid value and face value is £20 or more) is it fair to ask that person to buy a new railcard. If the railcard is simply forgotten for a journey, the ticket value increases to £90 before it's reasonable to ask the person to pay for another railcard.
(based on £30 a year being the cost of most railcards and there being a £10 admin fee for loss or theft)
The railway won't give a refund for purchasing a new card when you shouldn't need to... So why not give the opportunity to excess a railcard-discounted ticket to a non-railcard-discounted ticket, if requested before it is used?
I should add: It's obviously my opinion that there should be the opportunity for genuine mistakes/unfortunate circumstances to be resolved before travel. Whether that's the case or not isn't my decision, but the problem arises when some TOCs have different policies to others on the matter and nowhere is such information available - the same problem as most other problems on the National Rail network.