Maybe we should ban children, too. I find they are less well-behaved and noisier than people's pets are
I also agree with this idea.
You've basically stated all of the problems with dogs on trains right there - not once have I seen a dog on a train that was not either sitting on a seat or a table, or blocking the aisle by sitting/sleeping in the aisle or walking/running up and down the carriage.
I travel often with my dog, by train, he never causes any issues and he just goes to sleep on my lap, we never sit at a table, and he's never sat directly on a seat - he has on a few very empty trains slept on a seat, but I put my coat across the seat covering it entirely. He is always leashed, and I tend to carry him, but when he is walking along he is by my side, and is hyper focused on walking, he doesn't stop to sniff, or get distracted he will just walk.
I also try and get first class, so I can have a solo seat with nobody sat next to us, and he can just sleep on my lap. A lot of train crew will say hello to him, tell him he's being a good boy, and a few have sat and given him a little treat.
Personally I think it's utterly ridiculous that something an awful lot of people are afraid of, many people are allergic to, and that can often be violent are allowed on public transport.
Public transport should be accessible to all, and part of that is banning things that can be harmful or off putting to many people (like smoking/vaping), so allowing things that can cause issues with many people is going in the wrong direction for accessibility (even if some people are happy about it - in the same way, there are a minority that would like to be able to smoke on trains, but they aren't suddenly deciding to allow it).
So;
You want to ban dogs because 'an awful lot of people are afraid of' them but we're a nation of pet lovers, the vast majority of people own a pet of some kind.
You want to ban them due to peoples allergies, but these same people are travelling through a public space with a very high chance of encountering some kind of animal or an assistance animal (typically a dog, but can be other animals), and so the chances are if they're going to have a reaction then they'll already be taking medication and taking precautions.
You want to ban them because they're often violent. Does this mean we're going to be banning passengers on a Saturday evening where a not insignificant portion of passengers could easily be perceived as violent or on a razors edge to flipping and becoming violent?
Let's be real, anyone with a dog that hasn't been trained properly, isn't cared for properly, and is possibly violent/unpredictable would still be travelling by train because it's not like the owner will give a f*** and most staff will want to avoid the conflict with said individual, BTP aren't going to be in a rush to attend.