Have only Read OPs posts - I am gleaning that DS is ADHD, possible with comorbid issues such as ASD etc? I am also gathering that he hasn’t learnt to drive because he isn’t ready/confident/found a teacher who can help him learn in the light of his ADHD. ADHD in and of itself doesn’t limit him, but it may seriously undermine confidence. I am ADHD and couldn’t find the focus until I have children. My DD is simply not ready (too distractible and anxious behind the wheel) so stopped lessons after 8hrs.
I am also wondering whether DS may be in that weird limbo post degree because he doesn't have a job (and perhaps most of his peers from college do?) I’d suggest he speaks to his GP to explore whether he may be tackling with autism burnout and/or depression after leaving uni. The GP can prescribe medication and/or refer for talking therapy. I did not get the sense from your OPs that he has friends locally - did he make friends at Uni? Can he go and visit them? Is he in touch with them now he is home (Discord seems to be the way my kids interact with friends. Is he on that?). It may be that he is missing the structure of university, the goals that arise from coursework and exams, and that he is also lonely without people his own age around him.
I’d try and find him an ADHD mentor - there are a few organisations that work with people on the spectrum/with NDs who help them get work experience, practice interview technique, write their CVs, and apply for FT jobs. There was one in Surrey where I live that worked with my DD on this - I’d contact the national autism society (if this is applicable to your DS) or ADHD equivalents to see if they can direct you to such a service in your area?
It may also be worthwhile contacting your GP and enquiring whether they would do ‘shared care’. Not all do, but ours does, so they happily prescribe my DD’s meds and so we’ve been able to buy an annual prepayment certificate. We were paying £90-120 pcm before this. You can also contact his private clinical provider and see if they have ADHD counsellors and mentors to help at this stage (my DD’s private clinical provider does).
I’d sit him down and explain that his behaviour is completely unacceptable and that while your house is his home, he is welcomed to live in it ONLY if he abides by agreed shared values and rules. You can compromise by agreeing that you will not enter his room, in exchange though he has to keep it clean. He is not earning, so clearly not paying rent - however, he will be entitled to UC whilst he is job seeking, so I’d contact citizens advice to get guidance on what he should be claiming. As he is not paying rent, you are entitled to insist that he contributes to the smooth running of the house - he should be doing his own washing/laundry for example. He can stick the vacuum over and prepare one family meal a week (and clean up the kitchen). This is all what he should have been doing at uni, and it prepares him for independence.
As the parent of two ASD/ADHD kids I know that the ‘tell him to move out’ strategy is pointless. We’ll never kick our kids out until we know they are working, settled and can take care of themselves, but I assume he largely did this at uni, so you can definitely insists he snaps out of his current behaviour.