It's not only weird that the Hillises chose the Tiernsee but that they seemed unaware there was a school there at all. Juliet must surely have told Kay O'Hara all about it...
I suppose if Mrs Hillis senior chose the Tiernsee, and Frank agreed, Kay didn't have any good reason to object.
As to knowing about the school, perhaps she just didn't take in properly what Juliet told her. Or if she did know about it, she couldn't exactly say 'Oh, that's the Chalet School, I knew a girl at Holloway who went there.' Mrs Hillis would have wanted to know all about Juliet, and see the school and call on Madge.
Kay must have had some nasty moments when they met Jack and he invited them to visit; if she knew the Russells were Juliet's guardians, she must have wondered what she'd told them.
Frank Hillis was presumably a decent chap, since he was Jack's friend, so I imagine his mother was a decent person too. She might not have been at all impressed with Kay's behaviour.