There is absolutely nothing from a regulatory perspective to prevent an airline from offering standby seats if it wished. The question has to be "why would you?".
On EU261, the regulation only applies if you have a seat confirmed (Article 3(2)(a)). Up to the point where your seat is confirmed, then you effectively have no rights if you are denied boarding or the flight is disrupted. EU261 (or UK261 if you're now leaving a UK airport!) is not a barrier.
Also on this point, you cannot ask a passenger to exempt themselves from EU261 - for example, in return for a cheaper ticket. Article 15 of the regulation prevents you from doing that - basically if a passenger accepts sub-par [or no] compensation, they can go back later and claim their entitlement. You can't therefore sell cheaper tickets with an opt-out from EU261, even though many consider that would be a worthwhile option - basically to give the consumer the choice.
On the topic of commercial standby fares, they used to exist very widely - British Midland had them across its domestic network. Marvellous they were too. They were about £25 one-way all-in back in the early/mid 90s.
There were two main reasons for these disappearing. The first was the advent of low-cost carriers who made £25 one-way fares pretty common place - you didn't have to take a risk on your travel plans to get one. If you could book in advance and have certainty for £25, most did. The second was the inversion of the yield management principles led by the LCCs - bookings got far more expensive towards the date of departure and the best deals were "reserved" for those booking early. Put another way, if the full fare was £150 and you had a standby fare of £25, for every last-minute passenger who might have been prepared to pay £150 but instead bought a cheapie £25 standby, you have to carry five more passengers to make up for the "opportunity cost" - the lost revenue - from that first one. It never happens, so therefore the standby fares dilute overall last-minute yields.
I can't think you'll see these back any time soon, even though there is no regulatory blocker to prevent it. I just can't see why any airline would put itself to the bother of doing it in today's world. And I'm sure you'd also have to get passengers to sign some disclaimer to say that you'd properly explained the conditions to them at the point of sale, to avoid creating yet another bandwagon onto which the EU261 claims people would no doubt seek to jump. The claims culture has done so much damage to the airline industry - heartily encouraged by that gob***** at Which? - that it is no wonder that relations between airlines and passengers are difficult.