Ugh, massive bugbear of mine. And sadly - I briefly thought I might be able to squeeze the funds together for a studio flat rather than flatshare when I moved last autumn and looked online for possible flats, that's when I noticed it - it seems to become more and more common to just ask for a guarantor as default, even from people in full employment who meet the income threshold.
I think it's changed sometime in the last 5-10 years, didn't used to be a thing for the first decade of me renting in London.
When I just googled for a bit more info on this I found this "charming" landlord blog (sharpen your pitchforks, people, you will need them).
Sadly, many tenants struggle to get rental guarantors, while most landlords and agent won't accept tenants without one. Here are a few solutions that WORK!
www.propertyinvestmentproject.co.uk
I clicked on the comments which are quite interesting in as far as there are only a couple from the time the blog was written (2015), but then more and more people saying in the last couple of years "wtf is going on, I have been renting for x years/decades and now can't find a place without a guarantor", so it does seem a recent development on top of all the other insults and injuries of the UK rental market.
Not only, of course, do people generally have a limited pool of people who they could ask for such a massive favour, they also often have to be UK homeowning and not retired to qualify. And disclose a massive amount of personal and financial information. And as I understand it is also very difficult to get out of it again (you can't just do it for a fixed time, I don't think), so it is a massive commitment for the guarantor.
It might be worth looking into one of those rent guarantee insurance schemes that FridgeMagnet mentioned - of course, the parasitic scum factor is rather off-putting and they sound pretty scammy, but I think they are a thing.
Council offering a similar service sounds a much better option, so fingers crossed for you.
Having said all that, the landlord of the flat I ended up moving into didn't ask me for any references nor a deposit! (One of the flatmates had been here for a few years already, but she hasn't paid a deposit either and we have signed a normal formal joint tenancy agreement together between three of us).
That almost put me off as well for being so unusual.
Have you tried
OpenRent | Property To Rent From Private Landlords ? All private landlords, I think, and not going through agencies, which I find preferable in terms of being able to speak to the landlord directly and potentially develop some sort of rapport.
Wishing you the best of luck!