Just a quick caveat that I am not a mental health professional and this advice is based on my personal life experiences. I feel like how you approach this heavily depends on what you mean by "depressed". If he is only mildly depressed, because he doesn't have anything to do all day, the "force him to get his shit together and tell him to get a job or he has to move out" approach can be quite helpful because it will force him to get out the house and interact with people.

If he is very severely depressed, giving him an ultimatum like that could backfire because it could lead to him feeling overwhelmed and freezing up even more. I know that it can be incredibly frustrating when someone with severe depression appears to be refusing to do anything or just saying "no" to helpful suggestions. There can sometimes be this death spiral situation with severe depression where someone struggles to do basic things because of their depression, and then every time they struggle they interpret it as proof their depression is right and they are a useless failure, so they start avoiding the thing they "failed" at, which just makes them even more depressed in the long run.

If he is more severely depressed, I'd suggest more of a "baby steps" approach. Encourage him to do something that he's been avoiding in as easy a way as possible, e.g. cooking a basic meal once a week, making sure he leaves the house for a five minute walk once a day, cleaning a bit of his room that's messy. Once he has seen he can regularly succeed at one of those "easy" tasks, slowly add more or add something a little more challenging, like trying to volunteer once a week, being responsible for cleaning the house, meeting up with his friends regularly in person. Eventually build up to him getting a part time job, then a full time job, then moving out.

If you can convince him to seek mental health support I would also strongly encourage that. Meds aren't an instant fix and don't work for everyone, but I know way too many people that entirely avoid them due to online fearmongering. I reacted badly to the antidepressants I was put on, which just meant they didn't do anything to fix my symptoms and I got some rare physical side effects I didn't like. The online narrative of them being an "easy" way out or immediately turning you into an emotionless robot is a massive oversimplification, I had a bad experience and I'd still recommend other depressed people try a few to see if they help, because they help for most people.

Therapy is also really useful, although I know accessing it can be hard. The free NHS cognitive behavioural therapy sessions aren't always enough if you have a more complicated long term issue to solve, but they can still provide good basic coping skills. Finding a decent private therapist who specialises in the kind of issues your brother has could help for more long-term interventions, but I am aware that's not always financially possible.

Whether it's through therapy or not, I'd also make sure any long-standing issue that might be causing your brother to feel depressed gets addressed. All the mental health walks in the world will be of limited effectiveness if there is something actively ongoing that is making your brother feel miserable all the time. There are a lot of things that could potentially cause that, and they probably won't be the same as what turned out to be the issue for me. If you or your brother suspects something deeper might be going on, try and look into ways to fix that. Therapy works really well for this, but so does stuff like journalling, etc.