Unfair tho it may seem, it's not even really about being in receipt of benefits.
Its about affordability. Someone claiming UC will generally be on a relatively low income, or an income that is low for a family affording the costs of children.
Landlords, and their lenders, want tenants who can comfortably afford the rent, with no assistance. They want this because it means even if some thing bad happens, the tenant/tenants are more likely to have savings or a second income to fall bavk on, & won't immediately be skint and fall quickly behind on payments.
Benefit claimants by default are going to poorer & have less surplus to absorb extra costs or a month out of work etc. They are inherently riskier to lend to.
In the case of OP, it is fabulous that they work hard etc but £1500 a month is not a huge income for someone with a child in childcare. Even if it became illegal to refuse to rent due to receipt of benefits, it would be easy for estate agents to simply carry out affordability checks, eg checking rent is not more than 25% of wage etc & block on that basis.
This is precisely why social housing is crucial and we need more of it. There's a shortage of housing in some parts of the country and as such the private market can be choosy and will not meet the needs of those on lower incomes. The state must intervene & provide more housing.