That breaks the whole problem because you've removed the part where Albert says something that gives Bernard the information he needs to work out the date.

I don't think it's really a good critique of a logic problem to say, "but if you remove this extremely important part, it doesn't work anymore"!

The conversation would need to be:

Bernard: I don't know when Cheryl's birthday is

Albert: I don't know when Cheryl's birthday is, and I knew that Bernard didn't know.

Bernard: Then I know when Cheryl's birthday is now.

Albert: Then I also know when Cheryl's birthday is.

Then it works. You only have to assume that the parties are logical and tell the truth.