It is refreshing to see that you have actually done your research before asking the question.
Unfortunately, based on the information in the initial post, I do not believe that the caretaker qualifies for a ODW visa.
Au-pairs are a separate category of workers. And while you correctly pointed out a selection of points that favour its application to the ODW visa, I will point out why that fails.
https://www.gov.uk/au-pairs-employment-law/au-pairs
An au pair isn’t classed as a worker or an employee if most of the following apply:
they’re a foreign national living with a family in the UK
they’re an EU citizen or have entered the UK on a Youth Mobility visa or student visa
they’re here on a cultural exchange programme
they’ve got a signed letter of invitation from the host family that includes details of their stay, eg accommodation, living conditions, approximate working hours, free time, pocket money
they learn about British culture from the host family and share their own culture with them
they have their own private room in the house, provided free of charge
they eat their main meals with the host family, free of charge
they help with light housework and childcare for around 30 hours a week, including a couple of evenings babysitting
they get reasonable pocket money
they can attend English language classes at a local college in their spare time
they’re allowed time to study and can practise their English with the host family
they sometimes go on holiday with the host family and help look after the children
they can travel home to see their family during the year
If you notice, the crucial thing that distinguishes au-pairs from other household workers (even nannies, housekeepers and gardeners are excluded) are that they are here to learn English and British culture and primarily look after children. Note that the number of hours that they can work are only about 30 hours a week.
Also, note that the family does not sponsor au pairs, but that they are here on their own, either as an EU citizen or on a Youth Mobility visa or student visa.
Now onto the ODW visa. There is a reason that this site do not have a section for Tier 3 visas. They were never created. Tier 3 visas was supposed to cater for low-skilled migration. But after the A8 accession to the EU, there was a flood of low-skilled migration from within the EU (primarily from Poland) that depressed wages for low-skilled workers in the UK. That is why even ODW workers have to be paid the national minimum wage, else everybody will import cheaper workers from India and the Phillipines while they live in the UK.
I could find nothing on the gov.uk website that suggests that there are exceptions to the ODW visa.
Note that the ODW not only must be paid the minimum wage, but also that they can not work more than 48 hours a week (by the Working Time Directive of the EU) and must get atleast 25 days of holiday. That is to say, the ODW worker will get British employment rights and will be governed by British and European employment law.
Does your relative really want all that cost and overhead?
