As somebody who, amongst many many other things, has to get people of all ages (and actually, in the case of "feet on the seats" and reactions to it, it really is something which affects all age groups) to behave less antisocially on the railway and has had this delightful task to do for several years, I wonder if I could comment...
I do not in any way condone violence by anybody, nor even justifiable (whatever that could/would be, if it exists) physical force by the public, against somebody behaving antisocially on the railway in any way. But "feet on the seats" is not simply mildly irritating behaviour, as has previously been pointed out.
Aside from being a potential Byelaws breach, there is no way for a passenger or even most train crew to know the in-depth detail of when train seats are cleaned, and allowing somebody to simply add undue levels of dirt from their shoes to the seat for an indefinite time period is inconsiderate to everybody - spreading an unnecessary further risk of disease which is already heightened by being on public transport in the first place, and does not need further encouragement. This issue is also capable of reducing seating on busy lines, in turn, causing minor crowd control problems; it may also lead to intimidation of other passengers who would like a seat but don't feel able to ask; and so on. It also adds problems under what we call the "broken windows theory" (different names available), in which I am a strong believer. This states that the more littering, poor behaviour and small amounts of graffiti, for example, the higher the risk of behavioural problems spreading into more severe issues. I spend a tremendous amount of effort making sure my bit of the railway is a welcoming one, which is as free as possible from this sort of thing, and I can tell you it works; but it only works if you have zero tolerance for it springing up again.
Part of the problem is that some staff and over-enthusiastic members of the public have no idea of how to implement a "zero-tolerance" policy correctly. In my personal opinion, the older gentleman whose restraining actions are partially the subject of this thread has mistakenly believed that "zero-tolerance" means that violence is justified; it is not. Had this gentleman decided to report this to the guard, the guard had not been able to deal with the issue, and the BTP had become involved, I would consider this far more appropriate; zero-tolerance obviously means showing very visibly that you will not tolerate it, but uncontrolled scenarios where the culprits are restrained does not mean the railway is effectively showing it won't tolerate this behaviour anyway. It simply becomes a visceral, small-scale battle between parts of society.
As it so happens, in the specific case of people putting feet on seats, I rarely have a problem getting them to take them off; with the right tone of voice and an appropriate explanation if need be, most people will realise very readily that they need to show some consideration. In fact, there are a number of train journeys every week where my only face-to-face interaction with passengers is to ask them to take their feet of the seats - but it's remarkable how few show any resentment, and indeed some people thank you when they realise the error of their ways, and that you have shown leniency just by warning them politely! However, feet on the seats is also symptomatic of people who are drunk and disorderly, fare-dodging and "not caring about it", trespassing and many other things with which I can anecdotally associate it over the last few years. By asking, and getting a response of one sort or another, you can sometimes uncover a multitude of issues, and the behaviour of these kids makes me suspicious, if all that is reported is true.
Most young people are just like everybody else in that they will remove their feet when asked; they are also just like everybody else when I say that if they don't, it usually indicates further antisocial or illegal behaviour which heightens my professional responsibility to indicate zero-tolerance of the issues. I would also encourage the public to exercise zero-tolerance but not by violence, instead by reporting this to the closest employee of a railway authority (TOC/BTP). If people do not remove their feet when asked, it is harder to gauge the problem when it is a member of the public asking; but at best it is indicative of ingrained thoughtlessness and selfishness, and at worst a regular disregard for safety or responsibilities such as paying for a ticket. This is what I typically find from my actual front-line experience, and I have to say it is especially worrying amongst those of a fairly young age and at the start of some sort of independence. So a lesson does need to be taught, but it should just be mentally awkward, rather than physically painful; a war between individuals is not going to do anything, but impression of a proper railway culture is better. In other words, it's likely that if these kids were being this disrespectful and awkward, even to a member of the public with little authority, it is possibly symptomatic of a greater issue; zero-tolerance IS necessary; violence is not, but somebody with the correct authority needs to give a reprimand and if need to seek backup from those able to detain the children until they can be dealt with as far as the law allows. This is far better than an attempt at detention for no real gain.