"Going on benefits" and "getting housed" are two separate things.
If he's off sick he'll be on SSP. After a while (6 months?) his employer will terminate his employment on the grounds that he's too unwell to do the job. At that point he claims UC, including the sickness element. This is because while sleeping on sofas he has no rent to pay. If someone wants to give him a room as a lodger, he can claim UC whilst on SSP, for help with the rental costs. This will be granted because SSP is a low income (assuming no savings). There's a cap on the rental costs and its age dependant too. He's entitled to either the one bedroom rate or the shared house rate. Look up the LHA for his postcode, that will state the maximum rental help. If he's entitled to the one bedroom rate he can claim upto that even if he's actually living in a shared house.
Council tax single adult discount for council tax may apply when he gets housing (obviously not for shared housing). As may council tax reduction, which is means tested. These are claimed separately to anything else.
But if he's someone's lodger then he's not homeless. Until they decide they don't want a lodger any more and give him notice to leave then he's homeless again.
The council will only help with housing in limited circumstances, including homelessness. His illness will give some priority points on top of homelessness giving priority. They don't have to provide a HA property, they can house people in private rental too, it's pot luck really. He has as much right to apply to the council for housing assistance as anyone else does. They'll let him know what assistance they can offer, if any. Eg if he doesn't qualify as homeless he may still be allowed to join the housing register and bid on properties, he'd just have a lower priority and take years to get something.
He is unlikely to be accepted by the council as homeless if he voluntarily left his last home because they asked him to. "Eviction" is something only the courts can do, not the LL.
At any point he can claim PIP, which is not means tested. Whether he qualifies or not depends on whether he is disabled by his condition. It's possible to be too ill to work but not disabled.
He needs help with the debt. IVA. Interest frozen. Bankruptcy. Whatever is appropriate.
He hopefully is involved with GP and secondary MH services, if not, he sounds like he needs to be. There may also be charities who can offer support.
£6k savings starts to affect means tested benefits. £16k is the cut off point. Even with savings that disqualify him from income based benefits, he'll still qualify for a limited period of contribution based benefits because he's been paying NI.