For me the low mood and depression symptoms were caused by a mismatch of how I wanted to behave, vs how I actually behaved and not understanding why or feeling I had any control over it.
I knew that feeling I was useless and a waste of space sounded like depression but I was also really convinced that it was probably true, and isn't depression when you're having delusions about how crap you are? So I was confused. And nothing I ever did to try to help myself ever worked which was depressing.
Being diagnosed changed a lot of this because I instantly had answers for a lot of the "Why is this constantly happening???" frustrations and I literally had a piece of paper saying no, you're not broken, your brain just has a different OS and trying to use instructions for the neurotypical OS won't work as effectively.
I agree that children may have missed trauma but adults are usually aware of their trauma. I don't think it's the story for all cases of ADHD - there are popular psychologists who claim this and I don't think it's very helpful.
Medication can be a bit of a frustrating trial and error process, I'm titrating at the moment and it takes a while because ideally you spend a while at each dose to compare it. I found it helpful to learn a bit about the different kinds. I'll provide some links at the end.
But basically, it doesn't do a whole lot on its own, it can help smooth out some brain processes which can help you do things though. So it can help with the overwhelm and the stimulation-seeking (though I have found that 20mg methylphenidate seems to make me seek even more stimulation)
Forgetfulness maybe not, but it can help you be more effective at setting up things like reminders, putting keys in the right place every time etc. I probably am no better at not forgetting things, but I am better at setting things up in advance which helps me not forget, if that makes sense? Like I knew some library books were due back today, so I found them all at the weekend and put them in a bag ready to take back. I never would have done that before, if I even remembered they were due, I would have left it all until the last minute and then been running around the house stressed shouting at the DC and we would have missed the bus.
I think for things like task persistence, starting new habits, staying in contact with friends, organisation etc, these are all skills/habits which need to be built. They won't automatically get better with medication (maybe a little bit) but (in theory) you'll find that your efforts to build skills and habits will go better, things stick for longer, so over time you should be able to make progress on goals like these and it should stick. Personally I haven't found this yet, but I think these are slower goals and it's hard to judge the effectiveness after just a short time.
It can help to pick a specific goal and try to work towards it so that you can compare your efforts now, without medication, and with medication. For example housework is one I've been working on via the methods in the podcast A Slob Comes Clean.