Hey mallard Great thread. Though obviously not great having depression and all that entails... Sorry.
I think you have a brilliant idea there. I was looking for exactly this sort of person last year, as my lack of physical abilities with my auto-immune issues, plus isolation was increasing depression and anxiety. I had a range of family and friends around but everyone is SO busy these days & visits were sporadic. Like you I didn't really want to ask people to do specific things.
I started with a cleaner/helper (or 2) until I found the one I clicked with. A non-native English Speaker my age, spotted on a local Community website. I just wanted some help to sort the house out to begin with. I've now increased her hours to visiting 3 times a week and it's been brilliant, as she helps me out with all sorts of things. As she's paid (& I've become more comfortable asking her to do things) she's taken on more duties. Her English is coming on really well and my physical AND mental health has improved knowing I am not fighting these things alone. So, a win win here.
Like others have said, chase references. You are in a vulnerable position at home and you need to find the right person who you can trust, to fit in to your personal space. I did think about local Church communities, but that seemed a bit hypocritical. Turns out my helper is a committed church goer anyway so I've outsourced the praying to her too 
I wish you all the very best. Severe depression approaching our long winters (if you are) is not much fun. I have sacrificed other things to fund my help and company as it makes such a huge difference. Takes the pressure off those immediately around me too. 