You are wrong. The definitive judgement regarding homelessness and intentionality is Samuels v Birmingham City Council which if you care to read it is available on Bailii.
For ease of reference you may like to read the recap at 17.46
'Housing authorities will need to consider whether the applicant can afford the housing costs without being deprived of basic essentials such as food, clothing, heating, transport and other essentials specific to their circumstances. Housing costs should not be regarded as affordable if the applicant would be left with a residual income that is insufficient to meet these essential needs. Housing authorities may be guided by Universal Credit standard allowances when assessing the income that an applicant will require to meet essential needs aside from housing costs, but should ensure that the wishes, needs and circumstances of the applicant and their household are taken into account. ..'
So the circumstances are relevant as is the needs of the household. Bearing in mind the huge increases in heating costs, and food costs in the preceding12 months, it is entirely relevant that many households will be struggling to meet their rent costs and will be falling into arrears at this time. Therefore they are unlikely to be found intentionally homeless providing the LA properly investigates the circumstances as is their legal duty under the Housing Act 1996.
To make a blanket statement that a household which takes on a level of rent which they later find they cannot meet is intentionally homeless is a total misrepresentation of the law as it currently stands and so utterly ridiculous I'm flabbergasted.
This statement worryingly is made by someone who purports to be knowledgeable in this area of practice. They're not.
And while we're on the subject Johnson's government did not make it easier to evict tenants. The Coronavirus Act 2020 between April 2020 and September 2021 had a blanket ban on possession proceedings for rent arrears which following its lifting led to many landlords selling up and leaving the rental market. Other than this, there has been the introduction of more onerous regulations for the PRS since 2007 which has meant tenants have more protection not less.
Op should speak to Environmental Health and try to ensure the landlord is presented with a Hazard Notice which will force him to do repairs and it sounds like they are long overdue. By having a Hazard Notice served, it will also protect the Op from eviction for at least 6 months.
On the matter of possession proceedings for rent arrears, disrepair is a defence and to use this effectively if necessary, it will ideally require a report from EH.
Op try and keep your rent arrears below two months as your landlord will not have grounds for a mandatory possession order. Although as stated earlier they will be able to use a s21 providing you are near or at the end of the fixed term (unless your landlord gets a Hazard notice from EH.
Seek advice from your LA homelessness team and/or Shelter or your local law centre.