Exactly. It's basically her purpose in life, what she likes to do and takes pride in doing. She loves the family and wants to do everything she can to take care of them. A lot of grandmothers are the same way. People seem to forget that she was begging to be allowed to make Mike breakfast early in the movie. People like her like to stay busy, especially since she thought she lost Sam, so she wanted to dote on her surrogate family first and foremost.

In the movie, Alice insisted on serving the family first and then sitting down. So it really had nothing to do with the family. Alice seemed to want to serve them.

They should have had a pie fight in the end

Mike and Carol never had the pies either.

Here's my issue with that: every time one of the Bradys would say that Alice was "like family" I wanted to jump though the screen and say "Like family?! What do you mean 'like family'? She IS family, you morons!"
I just hated that even after all those years of love and loyalty, even when she was no longer in their employ, she was still (purposely or not) singled out as separate from the Bradys themselves. But, as a viewer who tends to see these characters as real people while watching, or, at least as representative of real people, I think they should have made it clear that she was a Brady, name or not! Alice was an honorary Brady! And, it sucks that they never came out and said it!! (>_<)
Please excuse typos/funny wording; I use speech-recognition that doesn't always recognize!

she actually had to do the grocery shopping.
you all are forgetting she couldn't drive.
OH THANK YOU GOD! THANK YOU SO
BLOODY
MUCH!!! Basil Fawlty

She sometimes went grocery shopping, with or without Alice. She did it alone in the Brady episode where Greg is grounded from driving. Here it was a setup for her to find out Greg had borrowed a friend's car, by running into the friend's mom at the store. And she had to take Alice to the store (or elsewhere) since Alice did not drive.

Belle du Jour!
Basil: Don't mention the war. I mentioned it once, but I think I got away with it.

Here's my question: what did Mrs. Brady do all day? Alice was cooking and cleaning. Carol went shopping (not grocery, but for new clothes) and attended PTA meetings, but surely that wasn't a full time thing. So what did she do all day when Mike was at work, the kids were at work and Alice was cooking and cleaning?

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Agreed, Alice was far more involved in those kids' lives than the majority of housekeepers' in America, that's for sure. She was basically a third parent, yet you are correct she was treated as "the help" during that moment and some others.
When Mike and Carol went to go watch Bobby race in the Indy 500, she just waltzes over to the Brady house to keep an eye on the place. Yet, when the house was moved, nobody bothered to leave her a message about it. Nobody really kept her in the loop on things, either. It really did become awkward towards the end there, especially when Alice decides to start a "day care" for the grandchildren, I guess giving her some usefulness again.