This info is great! So they probably need a logo portraying warmth and care, while also keeping it professional, but shouldn’t be too cold and corporate.
I personally start by experimenting with the vibes of different fonts and type layout, just copying the company name in illustrator a bunch of times and trying different fonts, weights, cases, layout (stacked/one line), etc.
Adobe fonts actually has a filter called friendly, I wonder if you can start with some of those?
Once I have a few options i would start playing around with options for the icon.
Just my personal process:)
Hope this helps…