Are you bidding? The system has changed since then and you won't just be offered a place by waiting until you're top of the list. You'll never be top of the list if you don't bid. If you're in London or another major city and aren't homeless or have children (minors) living with you though, it's fair to say you probably don't have any realistic chance of getting given anything.
OP I was born in London and had to leave there because I couldn't afford the rent, not even on a flat share. It's life. If your relative is on minimum wage, that's the same wherever you are in the country, so they're going to have to move somewhere rentals are cheaper.
The issues they've got are multiple. I'm guessing they're not earning much or are on benefits, so maybe don't pass landlords financial checks and so someone else in a better financial position is always picked. Landlord especially won't touch people on benefits. They know that if the rules change the person won't be able to afford the rent and will dig their heels in until evicted by bailiffs in order to get help from the council. The tenant may have no other realistic choice, but it's very expensive for the landlord so they'll always choose someone earning well if they can.
Their age is going to rule them out of any houseshares really, they might find an HMO that'll take them, it'll probably be filled with others who could be classed a society's misfits eg addicts, people out of prison, and other social misfits for whatever reason who've reached middle age (because that's what over 35 is, it's only older people who thinks 40s is "young" and 50s "isn't old") without sorting out their earnings potential, finding friends/a partner and sorting out their living circumstances to afford to rent a flat. It's no use looking at the nice flatshares filled with young people in good parts of town, your relative has no chance with that.
Not having a reference won't be helping. You're a relative, you don't really count and they fell out with their last relative-landlord, which is a big red flag. If anyone finds that out, they would be mad to rent to them. They're basically asking someone to take them on as an unknown quantity and what landlord is going to do that when they can choose someone with a good reference from current landlord operating as a business and earning enough to pay the rent.
Not having a guarantor is likely to be a problem if a low earner. Don't be guarantor for them, it's not just box ticking, you'll be liable to pay their rent if they don't and they've already proved happy to take advantage of you for 5yrs. If something goes wrong for them they're not going to compromise their lifestyle to keep paying rent when they know if they default you'll have to pay.
I'd tell them to go to Citizens Advice Bureau because there may be schemes for providing an official guarantor if you have no family to do it, this definitely exists in countries abroad so perhaps it does here.
Realistically I'd suggest saving up a few months income, which should be easy on minimal rent for 5yrs. Then quitting their job (I'm guessing working remotely isn't a option and neither would be travel costs halfway across the country for interviews) and taking a cheap Air B&B place in a cheaper town for rentals, living off the savings whilst applying for every job they can and contacting every employment agency in the surrounding areas and taking anything offered. Once they have an income finding a flat or lodgings in an HMO should be easier in a cheaper area.
Before everyone shouts at me for saying to quit work, I know it's a risky strategy, but so is subsisting solely on benefits (because they've lost their job due to hygiene/tiredness) whilst sleeping rough because you've kicked them out. That's a lot harder place to get life straightened out from IMO than moving to some place more affordable with a few months of savings and solid determination to secure a job and home.
If they're single without dependents or severe disabilities they're not walking out of your place in six months and straight into council grotty and temporary accommodation, never mind permanent housing. Even if they do eventually get into temporary accommodation, there'll be a gap where they're on the streets (or sofa surfing if they have friends maybe) first.
They're being far too passive about this situation. I expect they think you won't actually kick them out onto the streets, since you already proved happy to trash your own boundaries of "short term only" long ago. If relying on that is their plan, you're going to have them living with you forever unless you also become far less passive and take the drastic action of kicking them out. They haven't gone willingly in 5yrs, you need to accept they're not going to go willingly now either. Cheap rent, minimal chores and a soft touch relative for a landlord is no incentive to move out.
Yohe got 3 options. Ignore it, hope it goes away and they leave voluntarily, which will never happen and they'll be with you forever. Kick them out to live on the streets of necessary, which is likely their only chance of getting council help. Or the in-between option of going all out yourself to help secure somewhere for them, which is probably going to involve a lot of time and energy and probably a fair amount of money too (like providing them with a years rent upfront to offer a landlord, which you'll realistically never be repaid by resentful relative who doesn't want to move out). As well as duping potential landlords as to the true character of the prospective tenant or implying the rental is for both of you not just them etc to make them look a more attractive prospect.
Personally I wouldn't get involved. You've done your bit giving them 5 fricking years and they've taken the piss with your generosity, sorting nothing out. For me, they'd be out on their own now. You have to be able to live with whatever decision you make though.