Condolences to all of you.
Yes, you should - but in a warm, loving 'Let's get you on your feet' way.
Help him plan what to do and how to get it - a college course, work experience, volunteering, get him used to doing a full part of looking after the home/his own washing/cooking/cleaning, whether or not he did those before (we don't know whether your Mum was sick for a while or whether it was sudden, so he might not know how to do those things).
It sounds like he's feeling lost. Which is understandable. And it makes sense to me that he needs family to guide him in a way that your Mum might have done (or might not), had she not died.
It's still early days for somebody young to 'snap out of it and get a job' - IIRC the job centre doesn't expect somebody to get a job for six months after their parent dies, and they're not the nicest, kindest organisation in the world, after all. It could also be that whilst living with your brother might have been essential at the time, he's missing the warmth of your mum.
He might also need to see the GP to ensure that he doesn't become long term depressed; if he's prepared to move to you, it sounds as though he hasn't got a girlfriend/boyfriend or a group of mates to give him support. Bereavement counselling might help him.
Trouble is, whilst it's a bit of a stereotype, older brothers tend to be more of the doing, getting on with things and not necessarily able to do the showing love that I hope he had from your Mum. He might have been very much still her little boy, especially if he's the youngest by a fair way. (He might also be a bit of a spoiled wotsit, but that doesn't make the loss any less painful).
If he comes to you understanding that you love him and will help him, it would be easier than if he thinks he's just sofa surfing until you get fed up. He's homeless. Renting a houseshare probably feels impossible (and he won't have the credit history, deposit or anything like it to make this achievable right now, in any case).
Give him a home to grow up from.
(ETA: I was fully independent at that age, but that was by choice, not through somebody else's death - and I knew I had places to go if something went wrong, so I wasn't completely alone - even so, I expect that I would have made far better choices, had I made the initial decision to do it with a home and loving support behind me).