Hi StKaAllen,
Congratulations on your new mattress 
The foundations that have cardboard on top are usually made with more widely spaced slats inside and are usually lower quality as well. I agree that the cardboard compromises the ventilation of the mattress but with these types of foundations the slats are usually too far apart anyway for use without anything on top of them and the cardboard is usually just an inexpensive way to provide more even support for the mattress.
You can read a little more about the different types of mattress protectors and some of the tradeoffs involved in each type in post #89 here.
If you are looking for a mattress encasement which completely encloses the mattress (which I normally wouldn’t suggest unless there are specific issues involved that would indicate that this would be a better choice) then this thread and this post may also be helpful.
Phoenix