It is normal that they will first tell you the obvious advice that does not help.
They are expecting that when you find this advice does not help, you will go back and tell them so. They do not mean "try this, and if it doesn't work, you're doing it wrong and there's nothing we can help you with". They have to tell you the normal obvious advice because for some people they won't have tried this and they need to try it first because it doesn't make sense to launch into investigations and underlying causes for every parent who is having an issue managing their child's behaviour.
Also, as you know, impulsive and unempathetic behaviour at two is much more normal than it is at three. I think that it is worth returning to say "Nothing has changed; I'm still having the same problem".
Some aspects of some of your posts - the idea that there is nothing that can be done to help your son, the idea that his personality is what is driving his behaviour and it can't be changed, the fact you've had suicidal thoughts, whether you plan to act on them or not - all of this is pointing towards depression. Talking therapy may be able to be accessed through your HV for longer than the NHS offer, or through the GP if not, antidepressants may also be an option.
What I did not fit into my long post yesterday is the link that I saw about your son feeling posessive over items and also reacting when other children come on the climbing frame - I wonder if he sees the climbing frame as "his" and this is why he is acting out when other people touch it because he has an expectation that they obviously don't share and so they don't meet it.
Pouring water on a baby's face is not very nice but it is likely to be totally ordinary experimentation and curiosity.
You mentioned some struggles with DS1 taking the baby's toys and that you have to ignore DS2 while you deal with DS1 over this. This is going to sound counterproductive, but what if you did not stop him doing this? My thinking here is that this is a child who clearly struggles with the idea that other children will take his toys and that will be bad. And when he tries to protect what he sees as "his toys", he's getting told off and punished, which perhaps makes him feel even more anxiety about the idea that other people will take things from him. Remember that at three, he may not be able to make a distinction between things like "my favourite teddy that is just mine" and "my best ever toy car" and "my brother's rattle that looks interesting because it's new" and "the communal large play structure that belongs to everyone". Working on the concept of understanding personal vs communal belongings might be helpful here. You probably want to do it outside of a conflict moment and make it fun and positive.
What we did when DS3 was little and DS2 wanted to take every toy the baby had, we would suggest to him "if you want the toy DS3 has, can you find him something else to play with first?" We showed him that by offering DS3 another toy, the baby would usually take it (because babies have short attention spans and like shiny things) and drop the item that DS2 wanted. That meant DS2 was happy because he got his item, and DS3 was not screaming having had something removed. It helped teach DS2 that DS3 was a person with emotions and thoughts and expectations too.
Once DS3 got a bit older and started going after DS2's toys directly, we modified this and got DS2 to run and find a similar/alternative for DS3. Again, scaffolding a solution where DS2 gets to keep his toy and DS3 gets something too. And now that they are older and DS3 isn't so easily fobbed off, I've started pointing out to DS2 that if he just waits (usually for less than 5 minutes) DS3 will almost always give him a turn. Sometimes that waiting is enforced
but often he is pretty happy to come and sit on my lap and just have some company in the fact it's hard to wait. And it seems to be paying off because it's frequently less necessary to intervene at all - they are spontaneously deciding to share or thinking about which items can be offered to the other. It's making the gap between their relative abilities quite obvious because DS3 at 2 is about at the same level of sharing as DS2 is at 5. But that in some ways makes it easier to reinforce the behaviour that we like.