Milly wanted Hettie out of the way so she could have Sam to herself. She wanted to be the 'lady' of the manor. Why would she not root for Adare even help him out as best she could. If he managed to lure Hettie away, as Milly suspected, that would work in her favor.
Woman, man! That's the way it should be Tarzan. [Tarzan and his mate]
Initially Milly did not root for Adare because he was making her lose her control of the household. But when she saw the romance between Adare and Hettie, I think she did root for Adare, though not openly. She was only a servant after all, so there was not much she could do.
If Adare had managed to lure Hettie away, Hettie would still be in Sam's mind/heart. More importantly, given that this took place in the 19th century, Milly would never have been able to marry Sam. The only solution for Milly was for Hettie to die.
I can accept your view but I disagree and still think it's a flaw in the movie.
It's been a while since I saw the movie but as I recall Milly had already convinced Sam, and Hettie herself, that Hettie was a worthless, crazy, lush who couldn't do anything right. She had given up running home and, by all appearances, everyone believed her to be a complete embarrassment to Sam. They didn't want to associate with her; that's why a lot of people had stopped accepting invitations to their home.
Milly had already taken over the role of caretaker and the person on whom Sam could rely. He was an ambitious man who wanted to fit in and be accepted so if something had happened to Hettie he might have continued to love and care for her but he would have moved on and probably taken Milly as his wife (or not, but that's what Milly would have anticipated).
Once Adare came arrived killing her off no longer made sense; especially since she hadn't thought to do it before. Milly's best bet would have been to convince Hettie to leave with Adare and vice versa. They had a past. She could have, and I think she did, work on Sam to get him to see her in a negative light.
Woman, man! That's the way it should be Tarzan. [Tarzan and his mate]