In my textbook (Grammar In Use by Hewings), it says:
We can make a suggestion with Why not + verb or Why don't / doesn't... (but not Why do not / does not)
However, please look at this Ngram Viewer that says otherwise. The book I'm using was first published in 1999 and the third edition is in 2015. I don't think if it has connection with old-fashioned, obsolete, etc. Do people use this? At least when they want to emphasize?
"Why not" is an idiom used for making a suggestion, if it wasn't an idiom it would hardly be grammatical. It isn't a question. If you say "Why not go shopping", you are not expecting an answer.
"Why don't you" can also be used for making a suggestion, but this is also a regular question form. "Why don't you go shopping." can be interpreted in context as a suggestion, or a question, expecting a reply.
Now there is a little oddity. The position of "not" can either be contracted to "don't", or it must be placed next to main verb.
Why do you not go shopping?
That is probably intended as a regular question, but it could be a suggestion. The suggestion idiom is conversational, and in conversation you'd use the contraction "don't".
However, You can't say "Why do not you go shopping". This is true of other Wh questions. "What do you not like to eat?"/"What don't you like to eat?" (and not "What do not you like to eat".) "Why do not you..." is ungrammatical, and can't be used as a suggestion.
You can say either Why don't you go? or Why do you not go?. But you cannot say *Why do not you go?
