I don't think it terribly matters which way it is decided - what I think is most important is that it is unified and the same across all of the network. If it is acceptable to board a train with an Off-Peak ticket for which it is invalid at an unbarriered station, and be able to pay just the normal excess, even if an opportunity to purchase said excess existed before boarding the train, then I see no reason why the situation should be different at a barriered station.
It has never made much sense to me that you could, on lines like WCML where the difference between Off-Peak and Anytime can be near to 4× due to there being no Super Off-Peak for most flows, simply buy the cheapest ticket which isn't TOC restricted and then excess away any time or route restrictions onboard the train if you happen to be checked. It seems totally counterintuitive that there is no penalty for doing this onboard the train where an opportunity to purchase existed before travelling, when for many other situations where a similar or even lesser payment would be required to regularise the situation there is a high penalty to buying on board if you could have done so before boarding.
But to come back to your main question, I fundamentally see nothing wrong with either selling people excesses at the gate line or telling them to get them on-board. Obviously the latter would reduce revenue, as those who are currently told they must buy a brand new ticket and who currently just fork out for a new Anytime, would only pay the excess and only if they were checked onboard before getting off. The former option would increase barrier queues, which are plenty long enough for some trains at peak time, quite significantly.