In my English textbook, there is a conversation like this.

A: Can you make a cake?
B: Sorry, I'm terrible at baking. Why don't I buy a cake instead?

The textbook says "Why don't I ...?" is used when we offer to help. For example, "Why don't I look for the restaurant?"

I feel that "Why don't I" sounds weird. I've never used it before. I just use "Why don't we ...?" and "Why don't you ...?" When I make a suggestion or offer to help, I simply use "Can I ...?" or "Is it okay if I ...?"

Do native speakers use "Why don't I" this way?

Question

In my English textbook, there is a conversation like this.

A: Can you make a cake?
B: Sorry, I'm terrible at baking. Why don't I buy a cake instead?

The textbook says "Why don't I ...?" is used when we offer to help. For example, "Why don't I look for the restaurant?"

I feel that "Why don't I" sounds weird. I've never used it before. I just use "Why don't we ...?" and "Why don't you ...?" When I make a suggestion or offer to help, I simply use "Can I ...?" or "Is it okay if I ...?"

Do native speakers use "Why don't I" this way?

Answer

Yes, we do.  It is a common phrase.  The other two examples have a slightly different meaning.   
Hope this helps! :)

Answer

Yep, We do. I do.
Why don't I answer this for you.
It's even more an offer of help, if I do it.

Answer

I feel like it’s used when you’re trying to convince something you should do something but gently, so offering help is a good example. It’s sort of like saying “there’s no reason why I shouldn’t/couldn’t _____” instead of asking “should I do _____?”

“I’m having lots of trouble with this but I want to do it myself!”
“Ok, why don’t I get dinner started while you’re working on it.”

Answer

Yes we do use "Why don't I" like Why don't I just go home.