My first thought would be to get him out and go for a walk together. Maybe early morning or evening, if he is overweight, he might be self conscious. I have found myself in the past, that the less you leave the house, the more difficult it becomes - so choosing a time when it's less busy, and you being sensitive to any anxiety he feels about being seen, might help. My DCs don't like going for walks around our neighbourhood for this reason, and so I will drive somewhere they are less likely to see familiar faces, and they feel happier about this. (They haven't explicitly told me this, but I've worked it out.)
I would build up to any expectations of getting a job slowly, with an emphasis on his mental health first and foremost. I very much doubt he's in the right place to get a job at the moment - but he will get there eventually if you don't make "getting a job" the focus, I think.
I would want to encourage good habits. You could frame the idea of an early morning walk followed by a healthy breakfast together as you wanting to spend more time with him rather than you being at your wit's end at how lazy & unmotivated he is.
It's great that he has mowed the lawn.
He seems lost and in need of some TLC and gentle guidance in my opinion. Maybe if going for a walk is too much to start with, you could start with a healthy breakfast together, and do the dishes together. When my son was in a similar place to your son, I had no expectations that he had to talk, and I think he appreciated that, we would put music on & play draughts, or just eat and then watch TV together.
Good luck. I wish both of you all the best